Systems and methods for focusing beams with mode division multiplexing

ABSTRACT

A system for focusing an orbital angular momentum (OAM) multiplexed beam comprising OAM signal processing circuitry for generating an OAM multiplexed signal. The OAM multiplexed signal includes a plurality of data streams each having a unique orbital angular momentum applied thereto and multiplexed together within the OAM multiplexed signal. Each unique orbital angular momentum has a beam helicity value greater than l=2. An antenna array control circuit controls transmission of the multiplexed OAM signal from each of a plurality of antennas in an antenna array toward a focus point located below the ground as a transmission beam to cause the transmitted OAM multiplexed signals to converge at the focus point below the ground at substantially a same time to overcome a divergence of the transmitted plurality of OAM multiplexed signals caused by the beam helicity value of greater than l=2 for each of the unique orbital angular momentum.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.15/230,823, filed Aug. 8, 2016, entitled SYSTEMS AND METHODS FORFOCUSING BEAMS WITH MODE DIVISION MULTIPLEXING, which will issue as U.S.Pat. No. 10,193,611 on Jan. 29, 2019. U.S. patent application Ser. No.15/230,823 is a continuation of U.S. patent application Ser. No.14/731,191, filed Jun. 4, 2015, entitled SYSTEMS AND METHODS FORFOCUSING BEAMS WITH MODE DIVISION MULTIPLEXING, now U.S. Pat. No.9,413,448, issued Aug. 9, 2016 (Atty. Dkt. No. NXGN-32372), which claimsbenefit of U.S. Provisional Application No. 62/035,224, filed Aug. 8,2014, entitled FOCUSING APPROACH FOR OAM-BASED FREE-SPACE AND RF (Atty.Dkt. No. NXGN-32317), the specification of which is incorporated byreference herein in its entirety.

TECHNICAL FIELD

The following relates to orbital angular momentum based communication,and more particularly, to more tightly focusing a beam that has beenprocessed using orbital angular momentum signals.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding, reference is now made to thefollowing description taken in conjunction with the accompanyingDrawings in which:

FIG. 1 illustrates various techniques for increasing spectral efficiencywithin a transmitted signal;

FIG. 2 illustrates a particular technique for increasing spectralefficiency within a transmitted signal;

FIG. 3 illustrates a general overview of the manner for providingcommunication bandwidth between various communication protocolinterfaces;

FIG. 4 illustrates the manner for utilizing multiple level overlaymodulation with twisted pair/cable interfaces;

FIG. 5 illustrates a general block diagram for processing a plurality ofdata streams within an optical communication system;

FIG. 6 is a functional block diagram of a system for generating orbitalangular momentum within a communication system;

FIG. 7 is a functional block diagram of the orbital angular momentumsignal processing block of FIG. 6;

FIG. 8 is a functional block diagram illustrating the manner forremoving orbital angular momentum from a received signal including aplurality of data streams;

FIG. 9 illustrates a single wavelength having two quanti-spinpolarizations providing an infinite number of signals having variousorbital angular momentums associated therewith;

FIG. 10A illustrates an object with only a spin angular momentum;

FIG. 10B illustrates an object with an orbital angular momentum;

FIG. 10C illustrates a circularly polarized beam carrying spin angularmomentum;

FIG. 10D illustrates the phase structure of a light beam carrying anorbital angular momentum;

FIG. 11A illustrates a plane wave having only variations in the spinangular momentum;

FIG. 11B illustrates a signal having both spin and orbital angularmomentum applied thereto;

FIGS. 12A-12C illustrate various signals having different orbitalangular momentum applied thereto;

FIG. 12D illustrates a propagation of Poynting vectors for various Eigenmodes;

FIG. 12E illustrates a spiral phase plate;

FIG. 13 illustrates a multiple level overlay modulation system;

FIG. 14 illustrates a multiple level overlay demodulator;

FIG. 15 illustrates a multiple level overlay transmitter system;

FIG. 16 illustrates a multiple level overlay receiver system;

FIGS. 17A-17K illustrate representative multiple level overlay signalsand their respective spectral power densities;

FIG. 18 illustrates comparisons of multiple level overlay signals withinthe time and frequency domain;

FIG. 19 illustrates a spectral alignment of multiple level overlaysignals for differing bandwidths of signals;

FIG. 20 illustrates an alternative spectral alignment of multiple leveloverlay signals;

FIG. 21 illustrates a typical OAM multiplexing scheme;

FIG. 22 illustrates various manners for converting a Gaussian beam intoan OAM beam;

FIG. 23A illustrates a fabricated metasurface phase plate;

FIG. 23B illustrates a magnified structure of the metasurface phaseplate;

FIG. 23C illustrates an OAM beam generated using the phase plate withl=+1;

FIG. 24 illustrates the manner in which a q-plate can convert a leftcircularly polarized beam into a right circular polarization orvice-versa;

FIG. 25 illustrates the use of a laser resonator cavity for producing anOAM beam;

FIG. 26A illustrates a vortex beam generator;

FIG. 26B illustrates the manner in which gratings of a linear waveguideproduce a tiltwave by diffraction;

FIG. 26C illustrates an on-chip OAM generator;

FIG. 26D illustrates three OAM emitters fabricated on a single chip;

FIG. 27A illustrates spatial multiplexing using cascaded beam splitters;

FIG. 27B illustrated demultiplexing using cascaded beam splitters andconjugated spiral phase holograms;

FIG. 28 illustrates a log polar geometrical transformation based on OAMmultiplexing and demultiplexing;

FIG. 29A illustrates an OAM multiplexer/demultiplexer using a photonicintegrated circuit;

FIG. 29B illustrates simulated and experimentally generated OAM beamsusing the photonic integrated circuit;

FIG. 29C illustrates a conceptual view of the 3D integrated device forOAM multiplexing and demultiplexing;

FIG. 30A illustrates an intensity profile of generated OAM beams andtheir multiplexing;

FIG. 30B illustrates the optical spectrum of each channel after eachmultiplexing for the OAM beams of FIG. 10A;

FIG. 30C illustrates the recovered constellations of 16-QAM signalscarried on each OAM beam;

FIG. 31A illustrates the steps to produce 24 multiplex OAM beams;

FIG. 31B illustrates the optical spectrum of a WDM signal carrier on anOAM beam;

FIG. 32A illustrates a turbulence emulator;

FIG. 32B illustrates the measured power distribution of an OAM beamafter passing through turbulence with a different strength;

FIG. 33A illustrates how turbulence effects mitigation using adaptiveoptics;

FIG. 33B illustrates experimental results of distortion mitigation usingadaptive optics;

FIG. 34 illustrates a free-space optical data link using OAM;

FIG. 35A illustrates simulated spot sized of different orders of OAMbeams as a function of transmission distance for a 3 cm transmittedbeam;

FIG. 35B illustrates simulated power loss as a function of aperturesize;

FIG. 36A illustrates a perfectly aligned system between a transmitterand receiver;

FIG. 36B illustrates a system with lateral displacement of alignmentbetween a transmitter and receiver;

FIG. 36C illustrates a system with receiver angular error for alignmentbetween a transmitter and receiver;

FIG. 37A illustrates simulated power distribution among different OAMmodes with a function of lateral displacement;

FIG. 37B illustrates simulated power distribution among different OAMmodes as a function of receiver angular error;

FIG. 38 illustrates a free-space communication system;

FIG. 39 illustrates a block diagram of a free-space optics system usingorbital angular momentum and multi-level overlay modulation;

FIGS. 40A-40C illustrate the manner for multiplexing multiple datachannels into optical links to achieve higher data capacity;

FIG. 40D illustrates groups of concentric rings for a wavelength havingmultiple OAM valves;

FIG. 41 illustrates a WDM channel containing many orthogonal OAM beams;

FIG. 42 illustrates a node of a free-space optical system;

FIG. 43 illustrates a network of nodes within a free-space opticalsystem;

FIG. 44 illustrates a system for multiplexing between a free spacesignal and an RF signal;

FIG. 45 illustrates the manner in which beam divergence increases forhigher orbital angular momentum values;

FIG. 46 illustrates a block diagram of a system for generating a focusedOAM beam;

FIG. 47 illustrates an OAM focused beam used in a ground-penetratingapplication;

FIG. 48 illustrates a microwave/free-space system providing a focusedOAM beam to a fixed receiver;

FIG. 49 illustrates a multi-point broadcast of a focused OAM beam;

FIG. 50 illustrates a block diagram of an antenna array for providing afocused OAM beam;

FIG. 51 illustrates a point like radiator generating a beam relative toa lens axis;

FIG. 52 illustrates a radiator in a radiator plane generating an imagein the image plane;

FIG. 53 illustrates a radiator array consisting of a plurality ofradiating antennas;

FIG. 54 illustrates various pulses produced from the radiating antennasof FIG. 53;

FIG. 55 illustrates the derivation of energy density of a spherical wavefrom a radiator r over various distances;

FIGS. 56A-56C illustrate the energy of a reflected focused wave from aradiator r and reflector R;

FIG. 57 illustrates a scattering of a focused wave by a point-likescatterer;

FIG. 58 illustrates an improvement of angular resolution between anunfocused ground probing radar and focused ground probing radar;

FIG. 59 illustrates the manner in which Hermite Gaussian beams andLaguerre Gaussian beams diverge when transmitted from phased arrayantennas;

FIG. 60A illustrates beam divergence between a transmitting aperture anda receiving aperture;

FIG. 60B illustrates the use of a pair of lenses for reducing beamdivergence;

FIG. 61 illustrates a simulation model utilizing a pair of transmitterlenses;

FIG. 62A illustrates the relationship between aperture diameter,transmission distance and power loss decreases;

FIG. 62B illustrates the relationship between power loss and equivalentfocal lengths for a 1 km link;

FIG. 62C illustrates the relationship between the relationship betweenpower loss and equivalent focal links for a 10 km link;

FIG. 63A illustrates simulated SIR of OAM +3;

FIG. 63B illustrates simulated SIR when OAM signals are transmitted withlateral displacement;

FIG. 63C illustrates simulated SIR when OAM signals are transmitted withreceiver angular error and transmitter pointing error in a 1 kmOAM-based FSO link;

FIG. 64 illustrates a setup of an OAM-based FSO link using transmitterlenses;

FIG. 65 shows a comparison between simulated and experimental power lossof OAM +3 as a function of receiver aperture size;

FIGS. 66A and 66B show SIR of OAM +3 when OAM +1 and +3 are transmittedwith angular error and displacement; and

FIGS. 67A and 67B show bit error rate of OAM +3 when OAM +1, +3 aretransmitted with angular error and displacement.

DETAILED DESCRIPTION

Referring now to the drawings, wherein like reference numbers are usedherein to designate like elements throughout, the various views andembodiments of system and method for communication using orbital angularmomentum with modulation are illustrated and described, and otherpossible embodiments are described. The figures are not necessarilydrawn to scale, and in some instances the drawings have been exaggeratedand/or simplified in places for illustrative purposes only. One ofordinary skill in the art will appreciate the many possible applicationsand variations based on the following examples of possible embodiments.

Referring now to the drawings, and more particularly to FIG. 1, whereinthere is illustrated two manners for increasing spectral efficiency of aspectrum based system. In general, there are at least two different waysto increase spectral efficiency 102 of a spectrum based system. Theincrease may be brought about by signal processing techniques 104 in themodulation scheme or using multiple access technique. Additionally, thespectral efficiency can be increased by creating new Eigen channels 106within the electromagnetic propagation. These two techniques arecompletely independent of one another and innovations from one class canbe added to innovations from the second class. The benefits ofcombination are multiplicative not additive. Therefore, the combinationof these two techniques creates a further innovation.

Spectral efficiency 102 is a key driver of the efficiency of a spectrumbased system. The spectral efficiency 102 is defined in units ofbit/sec/hz and the higher the spectral efficiency, the better the moreefficient the system and the more valuable the system. This is becausespectral efficiency 102 can translate to a greater number of users,higher throughput, higher quality or some of each within acommunications system and all can be traded against each other.

Regarding techniques using signal processing techniques or multipleaccess techniques. These techniques in spectrum based communicationssystems include innovations such as TDMA, FDMA, CDMA, EVDO, GSM, WCDMA,HSPA and the most recent OFDM techniques used in 4G WIMAX and LTE.Almost all of these techniques use decades-old modulation techniquesbased on sinusoidal Eigen functions called QAM modulation. Within thesecond class of techniques involving the creation of new Eigen channels106, the innovations include diversity techniques including space andpolarization diversity as well as multiple input/multiple output (MIMO)where uncorrelated radio paths create independent Eigen channels andpropagation of electromagnetic waves.

Referring now to FIG. 2, the present spectrum based system configurationintroduces two techniques, one from the signal processing techniques 104category and one from the creation of new eigen channels 106 categorythat are entirely independent from each other. Their combinationprovides a unique manner to increase the spectral efficiency of an endto end spectrum based system from twisted pair and cable to fiberoptics, to free space optics, to RF used in cellular, backhaul andsatellite. The first technique involves the use of a new signalprocessing technique using new orthogonal signals to increase thespectral efficiency of QAM modulation by the introduction ofnon-sinusoidal functions. This improvement is referred to as quantumlevel overlay (QLO) 202. The second technique involves the applicationof new electromagnetic wavefronts using a property of electromagneticwaves or photon, called orbital angular momentum (QAM) 104 to similarlyincrease spectrum efficiency. These electromagnetic wavefronts canaccess the entire electromagnetic spectrum for radio frequencies throughvisible light and beyond. Application of each of these techniques 202and uniquely increases by orders of magnitude spectral efficiency 206within spectrum based systems. In one embodiment, the spectrum basedsystem includes spectrum based communications systems, but there areother embodiments such as radar that are not communications systems.

With respect to the quantum level overlay technique 202, new eigenfunctions are introduced that, when overlapped (on top of one anotherwithin a symbol), significantly increase the spectral efficiency of thesystem. The quantum level overlay technique 302 borrows from quantummechanics, special orthogonal signals that reduce the time bandwidthproduct and thereby increase the spectral efficiency of the channel.Each orthogonal signal is overlaid within the symbol acts as anindependent channel. These independent channels differentiate thetechnique from existing modulation techniques.

With respect to the application of orbital angular momentum 204, thistechnique introduces twisted electromagnetic waves, or light beams,having helical wave fronts that carry orbital angular momentum (OAM).Different OAM carrying waves/beams can be mutually orthogonal to eachother within the spatial domain, allowing the waves/beams to beefficiently multiplexed and demultiplexed within a link. OAM beams areinteresting in systems due to their potential ability to multiplexmultiple independent data carrying channels into a single frequency.

With respect to the combination of quantum level overlay techniques 202and orbital angular momentum application 204, the combination is uniqueas the OAM multiplexing technique is separate from, but compatible with,other electromagnetic techniques such as wave length and polarizationdivision multiplexing. Use of these two techniques together intoexisting electromagnetic systems further increases system performance.The application of these techniques together in a system can be used inany spectrum based system and in one embodiment, a communicationssystem, can materially increase the spectrum efficiency of said systemover twisted pair and cable to fiber optics, to free space optics, to RFused in cellular/backhaul and satellites.

Each of these techniques can be applied independent of one another, butthe combination provides a unique opportunity to not only increasespectral efficiency, but to increase spectral efficiency withoutsacrificing distance or signal to noise ratios.

The Shannon Capacity Equation, can be used to determine if spectralefficiency is increased in a system. Increased spectral efficiency canbe mathematically translated to more bandwidth. Since bandwidth has avalue, one can easily convert spectral efficiency gains to financialgains for the business impact of using higher spectral efficiency. Also,increased spectral efficiency allows sophisticated forward errorcorrection (FEC) techniques to be used, the net impact is higher qualitybut with the sacrifice of some bandwidth. However, if one can achievehigher spectral efficiency (or more virtual bandwidth), one cansacrifice some of the gained bandwidth for FEC and therefore higherspectral efficiency can also translate to higher quality.

Spectrum based system operators and their vendors are interested inincreasing spectral efficiency. However, the issue with respect to thisincrease is the corresponding cost of increasing spectral efficiency.Each technique at different layers of the system have a different pricetag associated therewith. Techniques that are implemented at a physicallayer have the most impact as all other techniques can be superimposedon top of the lower layer techniques and thus increase the spectralefficiency further. The price tag for some of the techniques can bedrastic when one considers other associated costs. For example, onemethod of increasing spectral efficiency, the multiple input multipleoutput (MIMO) technique, uses additional antennas to create additionalpaths where each RF path can be treated as an independent channel andthus increase the aggregate spectral efficiency. In the MIMO scenario,in addition to the costs of additional antennas and processing, theoperator has other associated soft costs dealing with MIMO such asantenna installation, coils, additional lease costs, costs to increasethe structural integrity of the antenna structure, etc. These techniquesnot only have tremendous cost, but they have huge timing issues as theseactivities take time and the achieving of higher spectral efficiencycomes with significant delays which can also be translated to financiallosses.

The quantum level overlay technique (QLO) 202 has an advantage that theindependent channels are created within the symbols without needing newantennas and also can be used in existing modulation systems. This willhave a tremendous cost and time benefit compared to other techniques.Also, the quantum layer overlay technique 202 is a physical layertechnique, which means that the other techniques at higher layers of theprotocol can receive the benefit of the QLO techniques 202 and thusincrease the spectral efficiency even further. QLO technique 202 usesstandard QAM modulation used in OFDM based multiple access technologiessuch as WIMAX or LTE. QLO technique 202 basically enhances the QAMmodulation at the transceiver by injecting new signals to the I & Qcomponents of the baseband and overlaying them before QAM modulation aswill be more fully described herein below. At the receiver, the reverseprocedure is used to separate the overlaid signals and the net effect isa pulse shaping that allows better localization of the spectrum comparedto standard QAM or even the root raised cosine. The impact of thistechnique is a significantly higher spectral efficiency.

Referring now more particularly to FIG. 3, there is illustrated ageneral overview of the manner for providing improved spectralefficiency within various communication protocol interfaces 302, using acombination of multiple level overlay modulation 304 and the applicationof orbital angular momentum 306 to increase the number of communicationschannels.

The various communication protocol interfaces 302 may be comprised of avariety of system links using the electromagnetic spectrum, such as RF,cable or twisted pair, or optical making use of light wavelengths suchas fiber-optic communications or free-space optics. Various types of RFcommunications may include a combination of RF microwave, RF satellitecommunication, nomadic and mobile wireless systems, as well asmultiplexing between RF and free-space optics in real time.

By combining a multiple layer overlay modulation technique 304 withorbital angular momentum (OAM) technique 306, a higher throughput overvarious types of system 302 may be achieved. The use of multiple leveloverlay modulation alone without OAM increases the spectral efficiencyof systems 302, whether wired, optical, or wireless. However, togetherwith OAM, the increase in spectral efficiency is even more significant.

Multiple overlay modulation techniques 304 provide a new degree offreedom beyond the conventional 2 degrees of freedom, with time T andfrequency F being independent variables in a two-dimensional notationalspace defining orthogonal axes in an information diagram. This comprisesa more general approach rather than modeling signals as fixed in eitherthe frequency or time domain. Previous modeling methods using fixed timeor fixed frequency are considered to be more limiting cases of thegeneral approach of using multiple level overlay modulation 304. Withinthe multiple level overlay modulation technique 304, signals may bedifferentiated in two-dimensional space rather than along a single axis.Thus, the information-carrying capacity and/or spectral efficiency of asystem may be determined by a number of signals which occupy differenttime and frequency coordinates and may be differentiated in a notationaltwo-dimensional space.

Within the notational two-dimensional space, minimization of the timebandwidth product, i.e., the area occupied by a signal in that space,enables denser packing, and thus, the use of more signals, with higherresulting information-carrying capacity and/or spectral efficiency,within a fixed bandwidth. Given the frequency bandwidth delta (Δf), agiven signal transmitted through it in minimum time Δt will have anenvelope described by certain time-bandwidth minimizing signals. Thetime-bandwidth products for these signals take the form;

ΔtΔf=½(2n+1)  (1)

where n is an integer ranging from 0 to infinity, denoting the order ofthe signal.

These signals form an orthogonal set of infinite elements, where eachhas a finite amount of energy. They are finite in both the time domainand the frequency domain, and can be detected from a mix of othersignals and noise through correlation, for example, by match filtering.Unlike other wavelets, these orthogonal signals have similar time andfrequency forms.

The orbital angular momentum process 306 provides a twist to wave frontsof the electromagnetic fields carrying the data stream that may enablethe transmission of multiple data streams on the same frequency,wavelength, or other signal-supporting mechanism. This will increase thebandwidth over a system by allowing a single frequency or wavelength tosupport multiple eigen channels, each of the individual channels havinga different orthogonal and independent orbital angular momentumassociated therewith.

In one embodiment, referring now to FIG. 4, there is illustrated afurther communication implementation technique using the above describedtechniques as twisted pairs or cables carry electrons (not photons).Rather than using each of the multiple level overlay modulation 304 andorbital angular momentum techniques 306, only the multiple level overlaymodulation 304 can be used in conjunction with a single wirelineinterface and, more particularly, a twisted pair communication link or acable communication link 402. The operation of the multiple leveloverlay modulation 404, is similar to that discussed previously withrespect to FIG. 3, but is used by itself without the use of orbitalangular momentum techniques 306, and is used with either a twisted paircommunication link or cable interface communication link 402.

Referring now to FIG. 5, there is illustrated a general block diagramfor processing a plurality of data streams 502 for transmission in anoptical communication system. The multiple data streams 502 are providedto the multi-layer overlay modulation circuitry 504 wherein the signalsare modulated using the multi-layer overlay modulation technique. Themodulated signals are provided to orbital angular momentum processingcircuitry 506 which applies a twist to each of the wave fronts beingtransmitted on the wavelengths of the optical communication channel. Thetwisted waves are transmitted through the optical interface 508 over anoptical communications link such as an optical fiber or free spaceoptics communication system. FIG. 5 may also illustrate an RF mechanismwherein the interface 508 would comprise and RF interface rather than anoptical interface.

Referring now more particularly to FIG. 6, there is illustrated afunctional block diagram of a system for generating the orbital angularmomentum “twist” within a communication system, such as that illustratedwith respect to FIG. 3, to provide a data stream that may be combinedwith multiple other data streams for transmission upon a same wavelengthor frequency. Multiple data streams 602 are provided to the transmissionprocessing circuitry 600. Each of the data streams 602 comprises, forexample, an end to end connection carrying a voice call or a packetconnection transmitting non-circuit switch packed data over a dataconnection. The multiple data streams 602 are processed bymodulator/demodulator circuitry 604. The modulator/demodulator circuitry604 modulates the received data stream 602 onto a wavelength orfrequency channel using a multiple level overlay modulation technique,as will be more fully described herein below. The communications linkmay comprise an optical fiber link, free-space optics link, RF microwavelink, RF satellite link, wired link (without the twist), etc.

The modulated data stream is provided to the orbital angular momentum(OAM) signal processing block 606. Each of the modulated data streamsfrom the modulator/demodulator 604 are provided a different orbitalangular momentum by the orbital angular momentum electromagnetic block606 such that each of the modulated data streams have a unique anddifferent orbital angular momentum associated therewith. Each of themodulated signals having an associated orbital angular momentum areprovided to an optical transmitter 608 that transmits each of themodulated data streams having a unique orbital angular momentum on asame wavelength. Each wavelength has a selected number of bandwidthslots B and may have its data transmission capability increase by afactor of the number of degrees of orbital angular momentum l that areprovided from the OAM electromagnetic block 606. The optical transmitter608 transmitting signals at a single wavelength could transmit B groupsof information. The optical transmitter 608 and OAM electromagneticblock 606 may transmit l×B groups of information according to theconfiguration described herein.

In a receiving mode, the optical transmitter 608 will have a wavelengthincluding multiple signals transmitted therein having different orbitalangular momentum signals embedded therein. The optical transmitter 608forwards these signals to the OAM signal processing block 606, whichseparates each of the signals having different orbital angular momentumand provides the separated signals to the demodulator circuitry 604. Thedemodulation process extracts the data streams 602 from the modulatedsignals and provides it at the receiving end using the multiple layeroverlay demodulation technique.

Referring now to FIG. 7, there is provided a more detailed functionaldescription of the OAM signal processing block 606. Each of the inputdata streams are provided to OAM circuitry 702. Each of the OAMcircuitry 702 provides a different orbital angular momentum to thereceived data stream. The different orbital angular momentums areachieved by applying different currents for the generation of thesignals that are being transmitted to create a particular orbitalangular momentum associated therewith. The orbital angular momentumprovided by each of the OAM circuitries 702 are unique to the datastream that is provided thereto. An infinite number of orbital angularmomentums may be applied to different input data streams using manydifferent currents. Each of the separately generated data streams areprovided to a signal combiner 704, which combines the signals onto awavelength for transmission from the transmitter 706.

Referring now to FIG. 8, there is illustrated an embodiment in which theOAM processing circuitry 606 may separate a received signal intomultiple data streams. The receiver 802 receives the combined OAMsignals on a single wavelength and provides this information to a signalseparator 804. The signal separator 804 separates each of the signalshaving different orbital angular momentums from the received wavelengthand provides the separated signals to OAM de-twisting circuitry 806. TheOAM de-twisting circuitry 806 removes the associated OAM twist from eachof the associated signals and provides the received modulated datastream for further processing. The signal separator 804 separates eachof the received signals that have had the orbital angular momentumremoved therefrom into individual received signals. The individuallyreceived signals are provided to the receiver 802 for demodulationusing, for example, multiple level overlay demodulation as will be morefully described herein below.

FIG. 9 illustrates in a manner in which a single wavelength orfrequency, having two quanti-spin polarizations may provide an infinitenumber of twists having various orbital angular momentums associatedtherewith. The l axis represents the various quantized orbital angularmomentum states which may be applied to a particular signal at aselected frequency or wavelength. The symbol omega (ω) represents thevarious frequencies to which the signals of differing orbital angularmomentum may be applied. The top grid 902 represents the potentiallyavailable signals for a left handed signal polarization, while thebottom grid 904 is for potentially available signals having right handedpolarization.

By applying different orbital angular momentum states to a signal at aparticular frequency or wavelength, a potentially infinite number ofstates may be provided at the frequency or wavelength. Thus, the stateat the frequency Δω or wavelength 906 in both the left handedpolarization plane 902 and the right handed polarization plane 904 canprovide an infinite number of signals at different orbital angularmomentum states Δl. Blocks 908 and 910 represent a particular signalhaving an orbital angular momentum Δl at a frequency Δω or wavelength inboth the right handed polarization plane 904 and left handedpolarization plane 910, respectively. By changing to a different orbitalangular momentum within the same frequency Δω or wavelength 906,different signals may also be transmitted. Each angular momentum statecorresponds to a different determined current level for transmissionfrom the optical transmitter. By estimating the equivalent current forgenerating a particular orbital angular momentum within the opticaldomain and applying this current for transmission of the signals, thetransmission of the signal may be achieved at a desired orbital angularmomentum state.

Thus, the illustration of FIG. 9, illustrates two possible angularmomentums, the spin angular momentum, and the orbital angular momentum.The spin version is manifested within the polarizations of macroscopicelectromagnetism, and has only left and right hand polarizations due toup and down spin directions. However, the orbital angular momentumindicates an infinite number of states that are quantized. The paths aremore than two and can theoretically be infinite through the quantizedorbital angular momentum levels.

It is well-known that the concept of linear momentum is usuallyassociated with objects moving in a straight line. The object could alsocarry angular momentum if it has a rotational motion, such as spinning(i.e., spin angular momentum (SAM) 1002), or orbiting around an axis1006 (i.e., OAM 1004), as shown in FIGS. 10A and 10B, respectively. Alight beam may also have rotational motion as it propagates. In paraxialapproximation, a light beam carries SAM 1002 if the electrical fieldrotates along the beam axis 1006 (i.e., circularly polarized light1005), and carries OAM 1004 if the wave vector spirals around the beamaxis 1006, leading to a helical phase front 1008, as shown in FIGS. 10Cand 10D. In its analytical expression, this helical phase front 1008 isusually related to a phase term of exp(ilθ) in the transverse plane,where θ refers to the angular coordinate, and l is an integer indicatingthe number of intertwined helices (i.e., the number of 2π phase shiftsalong the circle around the beam axis). l could be a positive, negativeinteger or zero, corresponding to clockwise, counterclockwise phasehelices or a Gaussian beam with no helix, respectively.

Two important concepts relating to OAM include:

1) OAM and polarization: As mentioned above, an OAM beam is manifestedas a beam with a helical phase front and therefore a twistingwavevector, while polarization states can only be connected to SAM 1002.A light beam carries SAM 1002 of ±h/2π (h is Plank's constant) perphoton if it is left or right circularly polarized, and carries no SAM1002 if it is linearly polarized. Although the SAM 1002 and OAM 1004 oflight can be coupled to each other under certain scenarios, they can beclearly distinguished for a paraxial light beam. Therefore, with theparaxial assumption, OAM 1004 and polarization can be considered as twoindependent properties of light.

2) OAM beam and Laguerre-Gaussian (LG) beam: In general, an OAM-carryingbeam could refer to any helically phased light beam, irrespective of itsradial distribution (although sometimes OAM could also be carried by anon-helically phased beam). LG beam is a special subset among allOAM-carrying beams, due to that the analytical expression of LG beamsare eigen-solutions of paraxial form of the wave equation in acylindrical coordinates. For an LG beam, both azimuthal and radialwavefront distributions are well defined, and are indicated by two indexnumbers, l and p, of which l has the same meaning as that of a generalOAM beam, and p refers to the radial nodes in the intensitydistribution. Mathematical expressions of LG beams form an orthogonaland complete basis in the spatial domain. In contrast, a general OAMbeam actually comprises a group of LG beams (each with the same l indexbut a different p index) due to the absence of radial definition. Theterm of “OAM beam” refers to all helically phased beams, and is used todistinguish from LG beams.

Using the orbital angular momentum state of the transmitted energysignals, physical information can be embedded within the radiationtransmitted by the signals. The Maxwell-Heaviside equations can berepresented as:

$\begin{matrix}{{{\nabla{\cdot E}} = \frac{\rho}{ɛ_{0}}}{{\nabla{\times E}} = {- \frac{\partial B}{\partial t}}}{{\nabla{\cdot B}} = 0}{{\nabla{\times B}} = {{ɛ_{0}\mu_{0}\frac{\partial E}{\partial t}} + {\mu_{0}{j\left( {t,x} \right)}}}}} & (2)\end{matrix}$

where ∇ is the del operator, E is the electric field intensity and B isthe magnetic flux density. Using these equations, one can derive 23symmetries/conserved quantities from Maxwell's original equations.However, there are only ten well-known conserved quantities and only afew of these are commercially used. Historically if Maxwell's equationswhere kept in their original quaternion forms, it would have been easierto see the symmetries/conserved quantities, but when they were modifiedto their present vectorial form by Heaviside, it became more difficultto see such inherent symmetries in Maxwell's equations.

The conserved quantities and the electromagnetic field can berepresented according to the conservation of system energy and theconservation of system linear momentum. Time symmetry, i.e. theconservation of system energy can be represented using Poynting'stheorem according to the equations:

$\begin{matrix}{{Hamiltonian}\mspace{14mu} \left( {{total}\mspace{14mu} {energy}} \right)} & \; \\{H = {{\sum\limits_{i}{m_{i}\gamma_{i}c^{2}}} + {\frac{ɛ_{0}}{2}{\int{d^{3}{x\left( {{E}^{2} + {c^{2}{B}^{2}}} \right)}}}}}} & \; \\{{conservation}\mspace{14mu} {of}\mspace{14mu} {energy}} & \; \\{{\frac{{dU}^{mech}}{dt} + \frac{{dp}^{em}}{dt} + {\oint_{s^{\prime}}{d^{2}x^{\prime}{\cdot S}}}} = 0} & \;\end{matrix}$

The space symmetry, i.e., the conservation of system linear momentumrepresenting the electromagnetic Doppler shift can be represented by theequations:

$\begin{matrix}{{linear}\mspace{14mu} {momentum}} & \; \\{p = {{\sum\limits_{i}{m_{i}\gamma_{i}v_{i}}} + {ɛ_{0}{\int{d^{3}{x\left( {E \times B} \right)}}}}}} & \; \\{{conservation}\mspace{14mu} {of}\mspace{14mu} {linear}\mspace{14mu} {momentum}} & \; \\{{\frac{{dp}^{mech}}{dt} + \frac{{dp}^{em}}{dt} + {\oint_{s^{\prime}}{d^{2}x^{\prime}{{\hat{n}}^{\prime} \cdot T}}}} = 0} & \;\end{matrix}^{\square}$

The conservation of system center of energy is represented by theequation:

$\begin{matrix}{R = {{\frac{1}{H}{\sum\limits_{i}{\left( {x_{i} - x_{0}} \right)m_{i}\gamma_{i}c^{2}}}} + {\frac{ɛ_{0}}{2H}{\int{d^{3}{x\left( {x - x_{0}} \right)}\left( {{E}^{2} + {c^{2}{B}^{2}}} \right)}}}}} & (3)\end{matrix}$

Similarly, the conservation of system angular momentum, which gives riseto the azimuthal Doppler shift is represented by the equation:

$\begin{matrix}{{conservation}\mspace{14mu} {of}\mspace{14mu} {angular}\mspace{14mu} {momentum}} & \; \\{{\frac{{dJ}^{mech}}{dt} + \frac{{dJ}^{em}}{dt} + {\oint_{s^{\prime}}{d^{2}x^{\prime}{{\hat{n}}^{\prime} \cdot M}}}} = 0} & \;\end{matrix}$

For radiation beams in free space, the EM field angular momentum J^(em)can be separated into two parts:

$\begin{matrix}{J^{em} = {{ɛ_{0}{\int_{V^{\prime}}^{\;}\; {d^{3}{x^{\prime}\left( {E \times A} \right)}}}} + {ɛ_{0}{\int_{V^{\prime}}^{\;}\mspace{7mu} {d^{3}x^{\prime}{E_{i}\left\lbrack {\left( {x^{\prime} - x_{0}} \right) \times \nabla} \right\rbrack}A_{i}}}}}} & (4)\end{matrix}$

For each singular Fourier mode in real valued representation:

$\begin{matrix}{J^{em} = {{{- i}\frac{ɛ_{0}}{2\omega}{\int_{V^{\prime}}^{\;}{d^{3}{x^{\prime}\left( {E^{*} \times E} \right)}}}} - {i\frac{ɛ_{0}}{2\omega}{\int_{V^{\prime}}\ {d^{3}x^{\prime}{E_{i}\left\lbrack {\left( {x^{\prime} - x_{0}} \right) \times \nabla} \right\rbrack}E_{i}}}}}} & (5)\end{matrix}$

The first part is the EM spin angular momentum S^(em), its classicalmanifestation is wave polarization. And the second part is the EMorbital angular momentum L^(em) its classical manifestation is wavehelicity. In general, both EM linear momentum P^(em), and EM angularmomentum J^(em)=L^(em)+S^(em) are radiated all the way to the far field.

By using Poynting theorem, the optical vorticity of the signals may bedetermined according to the optical velocity equation:

$\begin{matrix}{{continuity}\mspace{14mu} {equation}} & \; \\{{{\frac{\partial U}{\partial t} + {\nabla{\cdot S}}} = 0},} & \;\end{matrix}$

where S is the Poynting vector

S=¼(E×H*+E*×H),  (6)

and U is the energy density

U=¼(ε|E| ²+μ₀ |H| ²),  (7)

with E and H comprising the electric field and the magnetic field,respectively, and ε and μ₀ being the permittivity and the permeabilityof the medium, respectively. The optical vorticity V may then bedetermined by the curl of the optical velocity according to theequation:

$\begin{matrix}{V = {{\nabla{\times v_{opt}}} = {\nabla{\times \left( \frac{{E \times H^{*}} + {E^{*} \times H}}{{ɛ{E}^{2}} + {\mu_{0}{H}^{2}}} \right)}}}} & (8)\end{matrix}$

Referring now to FIGS. 11A and 11B, there is illustrated the manner inwhich a signal and its associated Poynting vector in a plane wavesituation. In the plane wave situation illustrated generally at 1102,the transmitted signal may take one of three configurations. When theelectric field vectors are in the same direction, a linear signal isprovided, as illustrated generally at 1104. Within a circularpolarization 1106, the electric field vectors rotate with the samemagnitude. Within the elliptical polarization 1108, the electric fieldvectors rotate but have differing magnitudes. The Poynting vectorremains in a constant direction for the signal configuration to FIG. 11Aand always perpendicular to the electric and magnetic fields. Referringnow to FIG. 11B, when a unique orbital angular momentum is applied to asignal as described here and above, the Poynting vector S 1110 willspiral about the direction of propagation of the signal. This spiral maybe varied in order to enable signals to be transmitted on the samefrequency as described herein.

FIGS. 12A-12C illustrate the differences in signals having differenthelicity (i.e., orbital angular momentums). Each of the spiralingPoynting vectors associated with the signals 1102, 1104, and 1106provide a different shaped signal. Signal 1202 has an orbital angularmomentum of +1, signal 1204 has an orbital angular momentum of +3, andsignal 1206 has an orbital angular momentum of −4. Each signal has adistinct angular momentum and associated Poynting vector enabling thesignal to be distinguished from other signals within a same frequency.This allows differing type of information to be combined on the samefrequency, since these signals are separately detectable and do notinterfere with each other (Eigen channels).

FIG. 12D illustrates the propagation of Poynting vectors for variousEigen modes. Each of the rings 1220 represents a different Eigen mode ortwist representing a different orbital angular momentum within the samefrequency. Each of these rings 1220 represents a different orthogonalchannel. Each of the Eigen modes has a Poynting vector 1222 associatedtherewith.

Topological charge may be multiplexed to the frequency for either linearor circular polarization. In case of linear polarizations, topologicalcharge would be multiplexed on vertical and horizontal polarization. Incase of circular polarization, topological charge would multiplex onleft hand and right hand circular polarizations. The topological chargeis another name for the helicity index “I” or the amount of twist or OAMapplied to the signal. The helicity index may be positive or negative.In RF, different topological charges can be created and muxed togetherand de-muxed to separate the topological charges.

The topological charges l s can be created using Spiral Phase Plates(SPPs) as shown in FIG. 11E using a proper material with specific indexof refraction and ability to machine shop or phase mask, hologramscreated of new materials or a new technique to create an RF version ofSpatial Light Modulator (SLM) that does the twist of the RF waves (asopposed to optical beams) by adjusting voltages on the device resultingin twisting of the RF waves with a specific topological charge. SpiralPhase plates can transform a RF plane wave (l=0) to a twisted RF wave ofa specific helicity (i.e. l=+1).

These embodiments can create cross talk and multipath interference.However, cross talk and multipath interference can be corrected using RFMultiple-Input-Multiple-Output (MIMO). In one embodiment, most of thechannel impairments can be detected using a control or pilot channel andbe corrected using algorithmic techniques (closed loop control system).However, other techniques can be used to eliminate these channelimpairments.

As described previously with respect to FIG. 5, each of the multipledata streams applied within the processing circuitry has a multiplelayer overlay modulation scheme applied thereto.

Referring now to FIG. 13, the reference number 1300 generally indicatesan embodiment of a quantum level overlay (QLO) modulation system,although it should be understood that the term QLO and the illustratedsystem 1300 are examples of embodiments. The QLO system may comprise onesuch as that disclosed in U.S. Pat. No. 8,503,546 entitled MultipleLayer Overlay Modulation which is incorporated herein by reference. Inone example, the modulation system 1300 would be implemented within themultiple level overlay modulation box 504 of FIG. 5. System 1300 takesas input an input data stream 1301 from a digital source 1302, which isseparated into three parallel, separate data streams, 1303A-1303C, oflogical 1s and 0s by input stage demultiplexer (DEMUX) 1004. Data stream1301 may represent a data file to be transferred, or an audio or videodata stream. It should be understood that a greater or lesser number ofseparated data streams may be used. In some of the embodiments, each ofthe separated data streams 1303A-1303C has a data rate of 1/N of theoriginal rate, where N is the number of parallel data streams. In theembodiment illustrated in FIG. 13, N is 3.

Each of the separated data streams 1303A-1303C is mapped to a quadratureamplitude modulation (QAM) symbol in an M-QAM constellation, forexample, 16 QAM or 64 QAM, by one of the QAM symbol mappers 1305A-C. TheQAM symbol mappers 1305A-C are coupled to respective outputs of DEMUX1304, and produced parallel in phase (I) 1306A, 1308A, and 1310A andquadrature phase (Q) 1306B, 1308B, and 1210B data streams at discretelevels. For example, in 64 QAM, each I and Q channel uses 8 discretelevels to transmit 3 bits per symbol. Each of the three I and Q pairs,1306A-1306B, 1308A-1308B, and 1310A-1310B, is used to weight the outputof the corresponding pair of function generators 1307A-1307B,1309A-1309B, and 1311A-1311B, which in some embodiments generate signalssuch as the modified Hermite polynomials described above and weightsthem based on the amplitude value of the input symbols. This provides 2Nweighted or modulated signals, each carrying a portion of the dataoriginally from income data stream 1301, and is in place of modulatingeach symbol in the I and Q pairs, 1306A-1306B, 1308A-1308B, and1310A-1310B with a raised cosine filter, as would be done for a priorart QAM system. In the illustrated embodiment, three signals are used,SH0, SH1, and SH2, which correspond to modifications of H0, H1, and H2,respectively, although it should be understood that different signalsmay be used in other embodiments.

The weighted signals are not subcarriers, but rather are sublayers of amodulated carrier, and are combined, superimposed in both frequency andtime, using summers 1312 and 1316, without mutual interference in eachof the I and Q dimensions, due to the signal orthogonality. Summers 1312and 1316 act as signal combiners to produce composite signals 1313 and1317. The weighted orthogonal signals are used for both I and Qchannels, which have been processed equivalently by system 1300, and aresummed before the QAM signal is transmitted. Therefore, although neworthogonal functions are used, some embodiments additionally use QAM fortransmission. Because of the tapering of the signals in the time domain,as will be shown in FIGS. 16A through 16K, the time domain waveform ofthe weighted signals will be confined to the duration of the symbols.Further, because of the tapering of the special signals and frequencydomain, the signal will also be confined to frequency domain, minimizinginterface with signals and adjacent channels.

The composite signals 1313 and 1317 are converted to analogue signals1315 and 1319 using digital to analogue converters 1314 and 1318, andare then used to modulate a carrier signal at the frequency of localoscillator (LO) 1320, using modulator 1321. Modulator 1321 comprisesmixers 1322 and 1324 coupled to DACs 1314 and 1318, respectively. Ninetydegree phase shifter 1323 converts the signals from LO 1320 into a Qcomponent of the carrier signal. The output of mixers 1322 and 1324 aresummed in summer 1325 to produce output signals 1326.

QLO can be used in a variety of systems using different transportmediums, such as wire, optical, and wireless, and may be used inconjunction with QAM. This is because QLO uses spectral overlay ofvarious signals, rather than spectral overlap. Spectral efficiency maybe increased by an order of magnitude, through extensions of availablespectral resources into multiple layers. The number of orthogonalsignals is increased from 2, cosine and sine, in the prior art, to anumber limited by the accuracy and jitter limits of generators used toproduce the orthogonal polynomials. However, as the accuracy and jitterlimits of oscillators are improving additional orthogonal systems willbe possible. QLQ can be used with any multiple access system to increaseits spectral efficiency. For example, QLO extends each of the I and Qdimensions of QAM to any multiple access techniques such as GSM, codedivision multiple access (CDMA), wide band CDMA (WCDMA), high speeddownlink packet access (HSPDA), evolution-data optimized (EV-DO),orthogonal frequency division multiplexing (OFDM), world-wideinteroperability for microwave access (WIMAX), and long term evolution(LTE) systems. QLO may be further used in conjunction with othermultiple access (MA) schemes such as frequency division duplexing (FDD),time division duplexing (TDD), frequency division multiple access(FDMA), and time division multiple access (TDMA). Overlaying individualorthogonal signals over the same frequency band allows creation of avirtual bandwidth wider than the physical bandwidth, thus adding a newdimension to signal processing. This modulation is applicable to anyphysical median, such as, twisted pair, cable, fiber optic, satellite,broadcast, free-space optics, and all types of wireless access. Themethod and system are compatible with many current and future multipleaccess systems, including EV-DO, UMB, WIMAX, WCDMA (with or without),multimedia broadcast multicast service (MBMS)/multiple input multipleoutput (MIMO), HSPA evolution, and LTE.

Referring now to FIG. 14, an QLO demodulator 1400 is illustrated,although it should be understood that the term QLO and the illustratedsystem 1400 are examples of embodiments. The modulator 1400 takes asinput an QLO signal 1226 which may be similar to output signal 1326 fromsystem 1300. Synchronizer 1427 extracts phase information, which isinput to local oscillator 1420 to maintain coherence so that themodulator 1421 can produce base band to analogue I signal 1415 and Qsignal 1419. The modulator 1421 comprises mixers 1422 and 1424, which,coupled to OL1420 through 90 degree phase shifter 1423. I signal 1415 isinput to each of signal filters 1407A, 1409A, and 1411A, and Q signal1419 is input to each of signal filters 1407B, 1409B, and 1411B. Sincethe orthogonal functions are known, they can be separated usingcorrelation or other techniques to recover the modulated data.Information in each of the I and Q signals 1415 and 1419 can beextracted from the overlapped functions which have been summed withineach of the symbols because the functions are orthogonal in acorrelative sense.

In some embodiments, signal filters 1407A-1407B, 1409A-1409B, and1411A-1411B use locally generated replicas of the polynomials as knownsignals in match filters. The outputs of the match filters are therecovered data bits, for example, equivalence of the QAM symbols1406A-1406B, 1408A-1408B, and 1410A-1410B of system 1400. Signal filters1407A-1407B, 1409A-1409B, and 1411A-1411B produce 2n streams of n, I,and Q signal pairs, which are input into demodulators 1428-1433.Demodulators 1428-1433 integrate the energy in their respective inputsignals to determine the value of the QAM symbol, and hence the logicalis and 0s data bit stream segment represented by the determined symbol.The outputs of the modulators 1428-1433 are then input into multiplexers(MUXs) 1405A-1405C to generate data streams 1403A-1403C. If system 1400is demodulating a signal from system 1300, data streams 1403A-1403Ccorrespond to data streams 1303A-1303C. Data streams 1403A-1403C aremultiplexed by MUX 1404 to generate data output stream 1401. In summary,QLO signals are overlayed (stacked) on top of one another on transmitterand separated on receiver.

QLO may be differentiated from CDMA or OFDM by the manner in whichorthogonality among signals is achieved. QLO signals are mutuallyorthogonal in both time and frequency domains, and can be overlaid inthe same symbol time bandwidth product. Orthogonality is attained by thecorrelation properties, for example, by least sum of squares, of theoverlaid signals. In comparison, CDMA uses orthogonal interleaving ordisplacement of signals in the time domain, whereas OFDM uses orthogonaldisplacement of signals in the frequency domain.

In communications system, spectral efficiency may be increased for achannel by assigning the same channel to multiple users. This isfeasible if individual user information is mapped to special orthogonalfunctions. CDMA systems overlap multiple user information and views timeintersymbol orthogonal code sequences to distinguish individual users,and OFDM assigns unique signals to each user, but which are notoverlaid, are only orthogonal in the frequency domain. Neither CDMA norOFDM increases bandwidth efficiency. CDMA uses more bandwidth than isnecessary to transmit data when the signal has a low signal to noiseratio (SNR). OFDM spreads data over many subcarriers to achieve superiorperformance in multipath radiofrequency environments. OFDM uses a cyclicprefix OFDM to mitigate multipath effects and a guard time to minimizeintersymbol interference (ISI), and each channel is mechanistically madeto behave as if the transmitted waveform is orthogonal. (Sync functionfor each subcarrier in frequency domain.) However, spectral efficiencymay also be increased for a channel by assigning the same channel tomultiple processes, input/output channel or one like.

In contrast, QLO uses a set of functions which effectively form analphabet that provides more usable channels in the same bandwidth,thereby enabling high spectral efficiency. Some embodiments of QLO donot require the use of cyclic prefixes or guard times, and therefore,outperforms OFDM in spectral efficiency, peak to average power ratio,power consumption, and requires fewer operations per bit. In addition,embodiments of QLO are more tolerant of amplifier nonlinearities thanare CDMA and OFDM systems.

FIG. 15 illustrates an embodiment of an QLO transmitter system 1500,which receives input data stream 1501. System 1500 represents amodulator/controller 1501, which incorporates equivalent functionalityof DEMUX 1304, QAM symbol mappers 1305A-C, function generators1307A-1307B, 1309A-1309B, and 1311A-1311B, and summers 1312 and 1316 ofsystem 1300, shown in FIG. 13. However, it should be understood thatmodulator/controller 1501 may use a greater or lesser quantity ofsignals than the three illustrated in system 1300. Modulator/controller1501 may comprise an application specific integrated circuit (ASIC), afield programmable gate array (FPGA), and/or other components, whetherdiscrete circuit elements or integrated into a single integrated circuit(IC) chip.

Modulator/controller 1501 is coupled to DACs 1504 and 1507,communicating a 10 bit I signal 1502 and a 10 bit Q signal 1505,respectively. In some embodiments, I signal 1502 and Q signal 1505correspond to composite signals 1313 and 1317 of system 1300. It shouldbe understood, however, that the 10 bit capacity of I signal 1502 and Qsignal 1505 is merely representative of an embodiment. As illustrated,modulator/controller 1501 also controls DACs 1504 and 1507 using controlsignals 1503 and 1506, respectively. In some embodiments, DACs 1504 and1507 each comprise an AD5433, complementary metal oxide semiconductor(CMOS) 10 bit current output DAC. In some embodiments, multiple controlsignals are sent to each of DACs 1504 and 1507.

DACs 1504 and 1507 output analogue signals 1315 and 1319 to quadraturemodulator 1321, which is coupled to LO 1320. The output of modulator1320 is illustrated as coupled to a transmitter 1508 to transmit datawirelessly, although in some embodiments, modulator 1321 may be coupledto a fiber-optic modem, a twisted pair, a coaxial cable, or othersuitable transmission media.

FIG. 16 illustrates an embodiment of an QLO receiver system 1600 capableof receiving and demodulating signals from system 1500. System 1600receives an input signal from a receiver 1608 that may comprise inputmedium, such as RF, wired or optical. The modulator 1421 driven by LO1420 converts the input to baseband I signal 1415 and Q signal 1419. Isignal 1415 and Q signal 1419 are input to analogue to digital converter(ADC) 1609.

ADC 1609 outputs 10 bit signal 1610 to demodulator/controller 1601 andreceives a control signal 1612 from demodulator/controller 1601.Demodulator/controller 1601 may comprise an application specificintegrated circuit (ASIC), a field programmable gate array (FPGA),and/or other components, whether discrete circuit elements or integratedinto a single integrated circuit (IC) chip. Demodulator/controller 1601correlates received signals with locally generated replicas of thesignal set used, in order to perform demodulation and identify thesymbols sent. Demodulator/controller 1601 also estimates frequencyerrors and recovers the data clock, which is used to read data from theADC 1609. The clock timing is sent back to ADC 1609 using control signal1612, enabling ADC 1609 to segment the digital I and Q signals 1415 and1419. In some embodiments, multiple control signals are sent bydemodulator/controller 1601 to ADC 1609. Demodulator/controller 1601also outputs data signal 1401.

Hermite polynomials are a classical orthogonal polynomial sequence,which are the Eigenstates of a quantum harmonic oscillator. Signalsbased on Hermite polynomials possess the minimal time-bandwidth productproperty described above, and may be used for embodiments of QLOsystems. However, it should be understood that other signals may also beused, for example orthogonal polynomials such as Jacobi polynomials,Gegenbauer polynomials, Legendre polynomials, Chebyshev polynomials, andLaguerre polynomials. Q-functions are another class of functions thatcan be employed as a basis for QLO signals.

In quantum mechanics, a coherent state is a state of a quantum harmonicoscillator whose dynamics most closely resemble the oscillating behaviorof a classical harmonic oscillator system. A squeezed coherent state isany state of the quantum mechanical Hilbert space, such that theuncertainty principle is saturated. That is, the product of thecorresponding two operators takes on its minimum value. In embodimentsof an QLO system, operators correspond to time and frequency domainswherein the time-bandwidth product of the signals is minimized. Thesqueezing property of the signals allows scaling in time and frequencydomain simultaneously, without losing mutual orthogonality among thesignals in each layer. This property enables flexible implementations ofQLO systems in various communications systems.

Because signals with different orders are mutually orthogonal, they canbe overlaid to increase the spectral efficiency of a communicationchannel. For example, when n=0, the optimal baseband signal will have atime-bandwidth product of ½, which is the Nyquist Inter-SymbolInterference (ISI) criteria for avoiding ISI. However, signals withtime-bandwidth products of 3/2, 5/2, 7/2, and higher, can be overlaid toincrease spectral efficiency.

An embodiment of an QLO system uses functions based on modified Hermitepolynomials, 4n, and are defined by:

$\begin{matrix}{{\Psi_{n}\left( {t,\xi} \right)} = {\frac{\left( {\tanh \; \xi} \right)^{n/2}}{2^{n/2}\left( {{n!}\cosh \; \xi} \right)^{1/2}}e^{\frac{1}{2}{t^{2}{\lbrack{1 - {\tanh \; \xi}}\rbrack}}}{H_{n}\left( \frac{1}{\sqrt{2\; \cosh \; \xi \; \sinh \; \xi}} \right)}}} & (9)\end{matrix}$

where t is time, and ξ is a bandwidth utilization parameter. Plots ofψ_(n) for n ranging from 0 to 9, along with their Fourier transforms(amplitude squared), are shown in FIGS. 5A-5K. The orthogonality ofdifferent orders of the functions may be verified by integrating:

∫∫ψ_(n)(t,ξ)ψ_(m)(t,ξ)dtdξ  (10)

The Hermite polynomial is defined by the contour integral:

$\begin{matrix}{{{H_{n}(z)} = {\frac{n!}{2\pi \; i}{\oint{e^{{- t^{2}} + {2t\; 2}}t^{{- n} - 1}{dt}}}}},} & (11)\end{matrix}$

where the contour encloses the origin and is traversed in acounterclockwise direction. Hermite polynomials are described inMathematical Methods for Physicists, by George Arfken, for example onpage 416, the disclosure of which is incorporated by reference.

FIGS. 17A-17K illustrate representative QLO signals and their respectivespectral power densities based on the modified Hermite polynomials ψ_(n)for n ranging from 0 to 9. FIG. 17A shows plots 1701 and 1704. Plot 1701comprises a curve 1727 representing ω₀ plotted against a time axis 1702and an amplitude axis 1703. As can be seen in plot 1701, curve 1727approximates a Gaussian curve. Plot 1704 comprises a curve 1737representing the power spectrum of ψ₀ plotted against a frequency axis1705 and a power axis 1706. As can be seen in plot 1704, curve 1737 alsoapproximates a Gaussian curve. Frequency domain curve 1707 is generatedusing a Fourier transform of time domain curve 1727. The units of timeand frequency on axis 1702 and 1705 are normalized for basebandanalysis, although it should be understood that since the time andfrequency units are related by the Fourier transform, a desired time orfrequency span in one domain dictates the units of the correspondingcurve in the other domain. For example, various embodiments of QLOsystems may communicate using symbol rates in the megahertz (MHz) orgigahertz (GHz) ranges and the non-0 duration of a symbol represented bycurve 1727, i.e., the time period at which curve 1727 is above 0 wouldbe compressed to the appropriate length calculated using the inverse ofthe desired symbol rate. For an available bandwidth in the megahertzrange, the non-0 duration of a time domain signal will be in themicrosecond range.

FIGS. 17B-17J show plots 1707-1724, with time domain curves 1728-1736representing ψ₁ through ψ₉, respectively, and their correspondingfrequency domain curves 1738-1746. As can be seen in FIGS. 17A-17J, thenumber of peaks in the time domain plots, whether positive or negative,corresponds to the number of peaks in the corresponding frequency domainplot. For example, in plot 1723 of FIG. 17J, time domain curve 1736 hasfive positive and five negative peaks. In corresponding plot 1724therefore, frequency domain curve 1746 has ten peaks.

FIG. 17K shows overlay plots 1725 and 1726, which overlay curves1727-1736 and 1737-1746, respectively. As indicated in plot 1725, thevarious time domain curves have different durations. However, in someembodiments, the non-zero durations of the time domain curves are ofsimilar lengths. For an QLO system, the number of signals usedrepresents the number of overlays and the improvement in spectralefficiency. It should be understood that, while ten signals aredisclosed in FIGS. 17A-17K, a greater or lesser quantity of signals maybe used, and that further, a different set of signals, rather than theψ_(n) signals plotted, may be used.

QLO signals used in a modulation layer have minimum time-bandwidthproducts, which enable improvements in spectral efficiency, and arequadratically integrable. This is accomplished by overlaying multipledemultiplexed parallel data streams, transmitting them simultaneouslywithin the same bandwidth. The key to successful separation of theoverlaid data streams at the receiver is that the signals used withineach symbols period are mutually orthogonal. QLO overlays orthogonalsignals within a single symbol period. This orthogonality prevents ISIand inter-carrier interference (ICI).

Because QLO works in the baseband layer of signal processing, and someembodiments use QAM architecture, conventional wireless techniques foroptimizing air interface, or wireless segments, to other layers of theprotocol stack will also work with QLO. Techniques such as channeldiversity, equalization, error correction coding, spread spectrum,interleaving and space-time encoding are applicable to QLO. For example,time diversity using a multipath-mitigating rake receiver can also beused with QLO. QLO provides an alternative for higher order QAM, whenchannel conditions are only suitable for low order QAM, such as infading channels. QLO can also be used with CDMA to extend the number oforthogonal channels by overcoming the Walsh code limitation of CDMA. QLOcan also be applied to each tone in an OFDM signal to increase thespectral efficiency of the OFDM systems.

Embodiments of QLO systems amplitude modulate a symbol envelope tocreate sub-envelopes, rather than sub-carriers. For data encoding, eachsub-envelope is independently modulated according to N-QAM, resulting ineach sub-envelope independently carrying information, unlike OFDM.Rather than spreading information over many sub-carriers, as is done inOFDM, for QLO, each sub-envelope of the carrier carries separateinformation. This information can be recovered due to the orthogonalityof the sub-envelopes defined with respect to the sum of squares overtheir duration and/or spectrum. Pulse train synchronization or temporalcode synchronization, as needed for CDMA, is not an issue, because QLOis transparent beyond the symbol level. QLO addresses modification ofthe symbol, but since CDMA and TDMA are spreading techniques of multiplesymbol sequences over time. QLO can be used along with CDMA and TDMA.

FIG. 18 illustrates a comparison of QLO signal widths in the time andfrequency domains. Time domain envelope representations 1801-1803 ofsignals SH0-SH3 are illustrated as all having a duration Ts. SH0-SH3 mayrepresent PSI₀-PSI₂, or may be other signals. The correspondingfrequency domain envelope representations are 1805-1807, respectively.SH0 has a bandwidth BW, SH1 has a bandwidth three times BW, and SH2 hasa bandwidth of 5 BW, which is five times as great as that of SH0. Thebandwidth used by an QLO system will be determined, at least in part, bythe widest bandwidth of any of the signals used. If each layer uses onlya single signal type within identical time windows, the spectrum willnot be fully utilized, because the lower order signals will use less ofthe available bandwidth than is used by the higher order signals.

FIG. 19 illustrates a spectral alignment of QLO signals that accountsfor the differing bandwidths of the signals, and makes spectral usagemore uniform, using SH0-SH3. Blocks 1901-1904 are frequency domainblocks of an OFDM signal with multiple subcarriers. Block 1903 isexpanded to show further detail. Block 1903 comprises a first layer 1903x comprised of multiple SH0 envelopes 1903 a-1903 o. A second layer 1903y of SH1 envelopes 1903 p-1903 t has one third the number of envelopesas the first layer. In the illustrated example, first layer 1903 x has15 SH0 envelopes, and second layer 1903 y has five SH1 envelopes. Thisis because, since the SH1 bandwidth envelope is three times as wide asthat of SH0, 15 SH0 envelopes occupy the same spectral width as five SH1envelopes. The third layer 1903 z of block 1903 comprises three SH2envelopes 1903 u-1903 w, because the SH2 envelope is five times thewidth of the SH0 envelope.

The total required bandwidth for such an implementation is a multiple ofthe least common multiple of the bandwidths of the QLO signals. In theillustrated example, the least common multiple of the bandwidth requiredfor SH0, SH1, and SH2 is 15 BW, which is a block in the frequencydomain. The OFDM-QLO signal can have multiple blocks, and the spectralefficiency of this illustrated implementation is proportional to(15+5+3)/15.

FIG. 20 illustrates another spectral alignment of QLO signals, which maybe used alternatively to alignment scheme shown in FIG. 18. In theembodiment illustrated in FIG. 20, the OFDM-QLO implementation stacksthe spectrum of SH0, SH1, and SH2 in such a way that the spectrum ineach layer is utilized uniformly. Layer 2000A comprises envelopes2001A-2001D, which includes both SH0 and SH2 envelopes. Similarly, layer2000C, comprising envelopes 2003A-2003D, includes both SH0 and SH2envelopes. Layer 2000B, however, comprising envelopes 2002A-2002D,includes only SH1 envelopes. Using the ratio of envelope sizes describedabove, it can be easily seen that BW+5 BW=3 BW+3 BW. Thus, for each SH0envelope in layer 2000A, there is one SH2 envelope also in layer 2000Cand two SH1 envelopes in layer 2000B.

Three Scenarios Compared:

1) QLO with 3 Layers defined by:

${{f_{0}(t)} = {W_{0}e^{- \frac{t^{2}}{4}}}},{W_{0} = 0.6316}$${{f_{1}(t)} = {W_{1}{te}^{- \frac{t^{2}}{4}}}},{W_{1} \approx 0.6316}$${{f_{2}(t)} = {{W_{2}\left( {t^{2} - 1} \right)}e^{- \frac{t^{2}}{4}}}},{W_{2} \approx 0.4466}$

(The current FPGA implementation uses the truncation interval of [−6,6].)2) Conventional scheme using rectangular pulse3) Conventional scheme using a square-root raised cosine (SRRC) pulsewith a roll-off factor of 0.5

For QLO pulses and SRRC pulse, the truncation interval is denoted by[−t1, t1] in the following figures. For simplicity, we used the QLOpulses defined above, which can be easily scaled in time to get thedesired time interval (say micro-seconds or nano-seconds). For the SRRCpulse, we fix the truncation interval of [−3T, 3T] where T is the symbolduration for all results presented in this document.

Application of OAM to Optical Communication

Utilization of OAM for optical communications is based on the fact thatcoaxially propagating light beams with different OAM states can beefficiently separated. This is certainly true for orthogonal modes suchas the LG beam. Interestingly, it is also true for general OAM beamswith cylindrical symmetry by relying only on the azimuthal phase.Considering any two OAM beams with an azimuthal index of l 1 and l 2,respectively:

U ₁(r,θ,z)=A ₁(r,z)exp(il ₁θ)  (12)

where r and z refers to the radial position and propagation distancerespectively, one can quickly conclude that these two beams areorthogonal in the sense that:

$\begin{matrix}{{\int\limits_{0}^{2\pi}{U_{1}U_{2}^{*}d\; \theta}} = \left\{ \begin{matrix}0 & {{{if}\mspace{14mu} _{1}} \neq _{2}} \\{A_{1}A_{2}^{*}} & {{{if}\mspace{14mu} _{1}} = _{2}}\end{matrix} \right.} & (13)\end{matrix}$

There are two different ways to take advantage of the distinctionbetween OAM beams with different l states in communications. In thefirst approach, N different OAM states can be encoded as N differentdata symbols representing “0”, “1”, . . . , “N−1”, respectively. Asequence of OAM states sent by the transmitter therefore represents datainformation. At the receiver, the data can be decoded by checking thereceived OAM state. This approach seems to be more favorable to thequantum communications community, since OAM could provide for theencoding of multiple bits (log 2(N)) per photon due to the infinitelycountable possibilities of the OAM states, and so could potentiallyachieve a higher photon efficiency. The encoding/decoding of OAM statescould also have some potential applications for on-chip interconnectionto increase computing speed or data capacity.

The second approach is to use each OAM beam as a different data carrierin an SDM (Spatial Division Multiplexing) system. For an SDM system, onecould use either a multi-core fiber/free space laser beam array so thatthe data channels in each core/laser beam are spatially separated, oruse a group of orthogonal mode sets to carry different data channels ina multi-mode fiber (MMF) or in free space. Greater than 1 petabit/s datatransmission in a multi-core fiber and up to 6 linearly polarized (LP)modes each with two polarizations in a single core multi-mode fiber hasbeen reported. Similar to the SDM using orthogonal modes, OAM beams withdifferent states can be spatially multiplexed and demultiplexed, therebyproviding independent data carriers in addition to wavelength andpolarization. Ideally, the orthogonality of OAM beams can be maintainedin transmission, which allows all the data channels to be separated andrecovered at the receiver. A typical embodiments of OAM multiplexing isconceptually depicted in FIG. 21. An obvious benefit of OAM multiplexingis the improvement in system spectral efficiency, since the samebandwidth can be reused for additional data channels.

OAM Beam Generation and Detection

Many approaches for creating OAM beams have been proposed anddemonstrated.

One could obtain a single or multiple OAM beams directly from the outputof a laser cavity, or by converting a fundamental Gaussian beam into anOAM beam outside a cavity. The converter could be a spiral phase plate,diffractive phase holograms, metal materials, cylindrical lens pairs,q-plates or fiber structures. There are also different ways to detect anOAM beam, such as using a converter that creates a conjugate helicalphase, or using a plasmonic detector.

Mode Conversion Approaches

Referring now to FIG. 22, among all external-cavity methods, perhaps themost straightforward one is to pass a Gaussian beam through a coaxiallyplaced spiral phase plate (SPP) 2202. An SPP 2202 is an optical elementwith a helical surface, as shown in FIG. 22A. To produce an OAM beamwith a state of l, the thickness profile of the plate should be machinedas lλθ/2π(n−1), where n is the refractive index of the medium. Alimitation of using an SPP 2202 is that each OAM state requires adifferent specific plate. As an alternative, reconfigurable diffractiveoptical elements, e.g., a pixelated spatial light modulator (SLM) 2204,or a digital micro-mirror device can be programmed to function as anyrefractive element of choice at a given wavelength. As mentioned above,a helical phase profile exp(ilθ) converts a linearly polarized Gaussianlaser beam into an OAM mode, whose wave front resembles an f-foldcorkscrew 2206, as shown at 2204. Importantly, the generated OAM beamcan be easily changed by simply updating the hologram loaded on the SLM2204. To spatially separate the phase-modulated beam from thezeroth-order non-phase-modulated reflection from the SLM, a linear phaseramp is added to helical phase code (i.e., a “fork”-like phase pattern2208 to produce a spatially distinct first-order diffracted OAM beam,carrying the desired charge. It should also be noted that theaforementioned methods produce OAM beams with only an azimuthal indexcontrol. To generate a pure LG_(l,p) mode, one must jointly control boththe phase and the intensity of the wavefront. This could be achievedusing a phase-only SLM with a more complex phase hologram.

Some novel material structures, such as metal-surface, can also be usedfor OAM generation. A compact metal-surface could be made into a phaseplate by manipulation of the structure caused spatial phase response. Asshown in FIGS. 23A and 23B, a V-shaped antenna array 2302 is fabricatedon the metal surface 2304, each of which is composed of two arms 2306,2308 connected at one end 2310. A light reflected by this plate wouldexperience a phase change ranging from 0 to 2π, determined by the lengthof the arms and angle between two arms. To generate an OAM beam, thesurface is divided into 8 sectors 2312, each of which introduces a phaseshift from 0 to 7π/4 with a step of π/4. The OAM beam with l=+1 isobtained after the reflection, as shown in FIG. 23C.

Referring now to FIG. 24, another interesting liquid crystal-baseddevice named “q-plate” 2402 is also used as a mode converter whichconverts a circularly polarized beam 2404 into an OAM beam 2406. Aq-plate is essentially a liquid crystal slab with a uniform birefringentphase retardation of it and a spatially varying transverse optical axis2408 pattern. Along the path circling once around the center of theplate, the optical axis of the distributed crystal elements may have anumber of rotations defined by the value of q. A circularly polarizedbeam 2404 passing through this plate 2402 would experience a helicalphase change of exp (ilθ) with l=2 q, as shown in FIG. 24.

Note that almost all the mode conversion approaches can also be used todetect an OAM beam. For example, an OAM beam can be converted back to aGaussian-like non-OAM beam if the helical phase front is removed, e.g.,by passing the OAM beam through a conjugate SPP or phase hologram.

Intra-Cavity Approaches

Referring now to FIG. 25, OAM beams are essentially higher order modesand can be directly generated from a laser resonator cavity. Theresonator 2500 supporting higher order modes usually produce the mixtureof multiple modes 2504, including the fundamental mode. In order toavoid the resonance of fundamental Gaussian mode, a typical approach isto place an intra-cavity element 2506 (spiral phase plate, tiled mirror)to force the oscillator to resonate on a specific OAM mode. Otherreported demonstrations include the use of an annular shaped beam aslaser pump, the use of thermal lensing, or by using a defect spot on oneof the resonator mirrors.

Referring now to FIG. 26A-26D, instead of using bulk free space optics,a more compact version of an OAM beam generator in the micrometer scaleis reported using a modified microring resonator 2602. The general ideais that whisper gallery mode (WGM) modes can be excited and confined ina ring resonator 2604. To change it into an OAM beam emitter, angulargrating structures 2606 are embedded into the regular ring resonator2604 to periodically vary the refractive index in the azimuthaldirection, as shown in FIG. 26A. The grating structure 2606 distributedalong the ring cavity is used to create diffractions on the guided modein the ring resonator 2604. The principle is similar to what a linearlydistributed grating does to the incoming light, as shown in FIG. 26B.The diffracted beam 2608 from the side of the ring resonator possesses ahelical phase front 2610 (FIG. 26C), the azimuthal state of which isdetermined by the difference between the azimuthal order of the modeguided in the ring and the period of grating elements in the ring 2604.The tuning of the E state of the generated beam can be achieved byexciting different orders of guided modes in the ring. A fast switchingbetween two different OAM states in the rate of 20 μs is demonstratedthrough the use of electrically contacted thermo-optical control. Due tothe miniaturized dimension, there is potential to produce an array ofOAM beam generators on a single photonic chip. A fabricated arrayincluding three identical ring resonators 2604 and their generatedoutput beams 2612 is shown in FIG. 26D at 2614, 2616 and 2618,respectively.

OAM Beams Multiplexing and Demultiplexing

One of the benefits of OAM is that multiple coaxially propagating OAMbeams with different l states provide additional data carriers as theycan be separated based only on the twisting wavefront. Hence, one of thecritical techniques is the efficient multiplexing/demultiplexing of OAMbeams of different l states, where each carries an independent datachannel and all beams can be transmitted and received using a singleaperture pair. Several multiplexing and demultiplexing techniques havebeen demonstrated, including the use of an inverse helical phasehologram to down-convert the OAM into a Gaussian beam, a mode sorter,free-space interferometers, a photonic integrated circuit, and q-plates.Some of these techniques are briefly described below.

Beam Splitter and Inverse Phase Hologram

A straightforward way of multiplexing is simply to use cascaded 3-dBbeam splitters (BS) 2702. Each BS 2702 can coaxially multiplex two beams2703 that are properly aligned, and cascaded N BSs can multiplex N+1independent OAM beams at most, as shown in FIG. 27A. Similarly, at thereceiver end, the multiplexed beam 2705 is divided into four copies 2704by BS 2702. To demultiplex the data channel on one of the beams (e.g.,with l=1_i), a phase hologram 2706 with a spiral charge of

−l

_i is applied to all the multiplexed beams 2704. As a result, thehelical phase on the target beam is removed, and this beam evolves intoa fundamental Gaussian beam, as shown in FIG. 27B. The down-convertedbeam can be isolated from the other beams, which still have helicalphase fronts by using a spatial mode filter 2708 (e.g., a single modefiber only couples the power of the fundamental Gaussian mode due to themode matching theory). Accordingly, each of the multiplexed beams 2704can be demultiplexed by changing the spiral phase hologram 2706.Although this method is very power-inefficient since the BSs 2702 andthe spatial mode filter 2706 cause a lot of power loss, it was used inthe initial lab demonstrations of OAM multiplexing/demultiplexing, dueto the simplicity of understanding and the reconfigurability provided byprogrammable SLMs.

Optical Geometrical Transformation-Based Mode Sorter

Referring now to FIG. 28, another method of multiplexing anddemultiplexing, which could be more power-efficient than the previousone (using beam splitters), is the use of an OAM mode sorter. This modesorter usually comprises three optical elements, including a transformer2802, a corrector 2804, and a lens 2806, as shown in FIG. 28. Thetransformer 2802 performs a geometrical transformation of the input beamfrom log-polar coordinates to Cartesian coordinates, such that theposition (x,y) in the input plane is mapped to a new position (u,v) inthe output plane, where

${u = {{- a}\; {\ln\left( \frac{\sqrt{x^{2} + y^{2}}}{b} \right)}}},$

and v=a arctan(y/x). Here, a and b are scaling constants. The corrector2804 compensates for phase errors and ensures that the transformed beamis collimated. Considering an input OAM beam with a ring-shaped beamprofile, it can be unfolded and mapped into a rectangular-shaped planewave with a tilted phase front. Similarly, multiple OAM beams havingdifferent 1 states will be transformed into a series of plane waves eachwith a different phase tilt. A lens 2806 focuses these tilted planewaves into spatially separated spots in the focal plane such that allthe OAM beams are simultaneously demultiplexed. As the transformation isreciprocal, if the mode sorter is used in reverse it can become amultiplexer for OAM. A Gaussian beam array placed in the focal plane ofthe lens 2806 is converted into superimposed plane waves with differenttilts. These beams then pass through the corrector and the transformersequentially to produce properly multiplexed OAM beams.

OAM Multiplexing/Demultiplexing Using Photonic Integrated Circuits

Integrated versions of an OAM (de)multiplexer can be performed usingplanar photonic waveguides 2902. The schematic concept of such a deviceis shown in FIG. 29A. A group of single mode waveguides interfaced withSMFs are placed in parallel as the input ports. The beam from each inputport is expanded to a plane wave with a phase tilt, and is then sampledby a number of path-length matched waveguides. The output apertures 2904of all the waveguides are circularly arranged. The coherent combinationof output beams from each aperture 2904 could evolve into an OAM beam,the state of which is determined by the position of the input port. Inprinciple, such a device with M waveguide apertures 2904 can support atmost N different OAM states ranging from −N/2 to N/2−1. Practically,much lower orders of OAM beams can be generated with a better quality.The simulated and experimentally observed OAM beams using a circuit with29 available apertures are shown in FIG. 29B. Note that instead of beingplaced perpendicular to the chip surface, the output aperture arrayscould also be arranged laterally in a 3-D structure, as shown in FIG.29C. This device was demonstrated and used in a free space opticalcommunication link multiplexed with two OAM states.

Free Space Communications

The first proof-of-concept experiment using OAM for free spacecommunications transmitted eight different OAM states each representinga data symbol one at a time. The azimuthal index of the transmitted OAMbeam is measured at the receiver using a phase hologram modulated with abinary grating. To effectively use this approach, fast switching isrequired between different OAM states to achieve a high data rate.Alternatively, classic communications using OAM states as data carrierscan be multiplexed at the transmitter, co-propagated through a freespace link, and demultiplexed at a receiver. The total data rate of afree space communication link has reached 100 Tbit/s or even beyond byusing OAM multiplexing. The propagation of OAM beams through a realenvironment (e.g., across a city) is also under investigation.

Basic Link Demonstrations

Referring now to FIGS. 30A-30C, initial demonstrates of using OAMmultiplexing for optical communications include free space links using aGaussian beam and an OAM beam encoded with OOK data. Four monochromaticGaussian beams each carrying an independent 50.8 Gbit/s (4×12.7 Gbit/s)16-QAM signal were prepared from an IQ modulator and free-spacecollimators. The beams were converted to OAM beams with l=−8, +10, +12and −14, respectively, using 4 SLMs each loaded with a helical phasehologram, as shown in FIG. 30A. After being coaxially multiplexed usingcascaded 3 dB-beam splitters, the beams were propagated through ˜1 mdistance in free-space under lab conditions. The OAM beams were detectedone at a time, using an inverse helical phase hologram and a fibercollimator together with a SMF. The 16-QAM data on each channel wassuccessfully recovered, and a spectral efficiency of 12.8 bit/s/Hz inthis data link was achieved, as shown in FIGS. 30B and 30C.

A following experiment doubled the spectral efficiency by adding thepolarization multiplexing into the OAM-multiplexed free-space data link.Four different OAM beams (f=+4, +8, −8, +16) on each of two orthogonalpolarizations (eight channels in total) were used to achieve a Terabit/stransmission link. The eight OAM beams were multiplexed anddemultiplexed using the same approach as mentioned above. The measuredcrosstalk among channels carried by the eight OAM beams is shown inTable 1, with the largest crosstalk being ˜−18.5 dB. Each of the beamswas encoded with a 42.8 Gbaud 16-QAM signal, allowing a total capacityof ˜1.4 (42.8×4×4×2) Tbit/s.

TABLE 1 OAM + 4 OAM + 8 OAM − 8 OAM + 16 Measured Crosstalk X-Pol.Y-Pol. X-Pol. Y-Pol. X-Pol. Y-Pol. X-Pol. Y-Pol. OAM + 4 (dB) X-Pol.−23.2 −26.7 −30.8 −30.5 −27.7 −24.6 −30.1 Y-Pol. −25.7 OAM + 8 (dB)X-Pol. −26.6 −23.5 −21.6 −18.9 −25.4 −23.9 −28.8 Y-Pol. −25.0 OAM − 8(dB) X-Pol. −27.5 −33.9 −27.6 −30.8 −20.5 26.5 −21.6 Y-Pol. −26.8 OAM +16 (dB) X-Pol. −24.5 −31.2 −23.7 −23.3 −25.8 −26.1 −30.2 Y-Pol. −24.0Total from other OAMs * (dB) −21.8 −21.0 −21.2 −21.4 −18.5 −21.2 −22.2−20.7

The capacity of the free-space data link was further increased to 100Tbit/s by combining OAM multiplexing with PDM (phase divisionmultiplexing) and WDM (wave division multiplexing). In this experiment,24 OAM beams (l=±4, ±7, ±10, ±13, ±16, and ±19, each with twopolarizations) were prepared using 2 SLMs, the procedures for which areshown in FIG. 31 at 3102-3106. Specifically, one SLM generated asuperposition of OAM beams with l=+4, +10, and +16, while the other SLMgenerated another set of three OAM beams with l=+7, +13, and +19 (FIG.31A). These two outputs were multiplexed together using a beam splitter,thereby multiplexing six OAM beams: l=+4, +7, +10, +13, +16, and +19(FIG. 31A). Secondly, the six multiplexed OAM beams were split into twocopies. One copy was reflected five times by three mirrors and two beamsplitters, to create another six OAM beams with inverse charges (FIG.31B). There was a differential delay between the two light paths tode-correlate the data. These two copies were then combined again toachieve 12 multiplexed OAM beams with l=±4, ±7, ±10, ±13, ±16, and ±19(FIG. 31B). These 12 OAM beams were split again via a beam splitter. Oneof these was polarization-rotated by 90 degrees, delayed by ˜33 symbols,and then recombined with the other copy using a polarization beamsplitter (PBS), finally multiplexing 24 OAM beams (with l=±4, ±7, ±10,±13, ±16, and ±19 on two polarizations). Each of the beam carried a WDMsignal comprising 100 GHz-spaced 42 wavelengths (1,536.34-1,568.5 nm),each of which was modulated with 100 Gbit/s QPSK data. The observedoptical spectrum of the WDM signal carried on one of the demultiplexedOAM beams (l=+10).

Atmospheric Turbulence Effects on OAM Beams

One of the critical challenges for a practical free-space opticalcommunication system using OAM multiplexing is atmospheric turbulence.It is known that inhomogeneities in the temperature and pressure of theatmosphere lead to random variations in the refractive index along thetransmission path, and can easily distort the phase front of a lightbeam. This could be particularly important for OAM communications, sincethe separation of multiplexed OAM beams relies on the helicalphase-front. As predicted by simulations in the literature, theserefractive index inhomogeneities may cause inter-modal crosstalk amongdata channels with different OAM states.

The effect of atmospheric turbulence is also experimentally evaluated.For the convenience of estimating the turbulence strength, one approachis to emulate the turbulence in the lab using an SLM or a rotating phaseplate. FIG. 32A illustrates an emulator built using a thin phase screenplate 3202 that is mounted on a rotation stage 3204 and placed in themiddle of the optical path. The pseudo-random phase distributionmachined on the plate 3202 obeys Kolmogorov spectrum statistics, whichare usually characterized by a specific effective Fried coherence lengthr0. The strength of the simulated turbulence 3206 can be varied eitherby changing to a plate 3202 with a different r0, or by adjusting thesize of the beam that is incident on the plate. The resultant turbulenceeffect is mainly evaluated by measuring the power of the distorted beamdistributed to each OAM mode using an OAM mode sorter. It was foundthat, as the turbulence strength increases, the power of the transmittedOAM mode would leak to neighboring modes and tend to be equallydistributed among modes for stronger turbulence. As an example, FIG. 32Bshows the measured average power (normalized) l=3 beam under differentemulated turbulence conditions. It can be seen that the majority of thepower is still in the transmitted OAM mode 3208 under weak turbulence,but it spreads to neighboring modes as the turbulence strengthincreases.

Turbulence Effects Mitigation Techniques

One approach to mitigate the effects of atmospheric turbulence on OAMbeams is to use an adaptive optical (AO) system. The general idea of anAO system is to measure the phase front of the distorted beam first,based on which an error correction pattern can be produced and can beapplied onto the beam transmitter to undo the distortion. As for OAMbeams with helical phase fronts, it is challenging to directly measurethe phase front using typical wavefront sensors due to the phasesingularity. A modified AO system can overcome this problem by sending aGaussian beam as a probe beam to sense the distortion, as shown in FIG.33A. Due to the fact that turbulence is almost independent of the lightpolarization, the probe beam is orthogonally polarized as compared toall other beams for the sake of convenient separation at beam separator3302. The correction phase pattern can be derived based on the probebeam distortion that is directly measured by a wavefront sensor 3204. Itis noted that this phase correction pattern can be used tosimultaneously compensate multiple coaxially propagating OAM beams. FIG.33 at 3310-3320 illustrate the intensity profiles of OAM beams with l=1,5 and 9, respectively, for a random turbulence realization with andwithout mitigation. From the far-field images, one can see that thedistorted OAM beams (upper), up to l=9, were partially corrected, andthe measured power distribution also indicates that the channelcrosstalk can be reduced.

Another approach for combating turbulence effects is to partially movethe complexity of optical setup into the electrical domain, and usedigital signal processing (DSP) to mitigate the channel crosstalk. Atypical DSP method is the multiple-input-multiple-output (MIMO)equalization, which is able to blindly estimate the channel crosstalkand cancel the interference. The implementation of a 4×4 adaptive MIMOequalizer in a four-channel OAM multiplexed free space optical linkusing heterodyne detection may be used. Four OAM beams (l=+2, +4, +6 and+8), each carrying 20 Gbit/s QPSK data, were collinearly multiplexed andpropagated through a weak turbulence emulated by the rotating phaseplate under laboratory condition to introduce distortions. Afterdemultiplexing, four channels were coherently detected and recordedsimultaneously. The standard constant modulus algorithm is employed inaddition to the standard procedures of coherent detection to equalizethe channel interference. Results indicate that MIMO equalization couldbe helpful to mitigate the crosstalk caused by either turbulence orimperfect mode generation/detection, and improve both error vectormagnitude (EVM) and the bit-error-rate (BER) of the signal in anOAM-multiplexed communication link. MIMO DSP may not be universallyuseful as outage could happen in some scenarios involving free spacedata links. For example, the majority power of the transmitted OAM beamsmay be transferred to other OAM states under a strong turbulence withoutbeing detected, in which case MIMO would not help to improve the systemperformance.

OAM Free Space Link Design Considerations

To date, most of the experimental demonstrations of opticalcommunication links using OAM beams took place in the lab conditions.There is a possibility that OAM beams may also be used in a free spaceoptical communication link with longer distances. To design such a datalink using OAM multiplexing, several important issues such as beamdivergence, aperture size and misalignment of two transmitter andreceiver, need to be resolved. To study how those parameters affect theperformance of an OAM multiplexed system, a simulation model wasdescribed by Xie et al, the schematic setup of which is shown in FIG.34. Each of the different collimated Gaussian beams 3402 at the samewavelength is followed by a spiral phase plate 3404 with a unique orderto convert the Gaussian beam into a data-carrying OAM beam. Differentorders of OAM beams are then multiplexed at multiplexor 3406 to form aconcentric-ring-shape and coaxially propagate from transmitter 3408through free space to the receiver aperture located at a certainpropagation distance. Propagation of multiplexed OAM beams isnumerically propagated using the Kirchhoff-Fresnel diffraction integral.To investigate the signal power and crosstalk effect on neighboring OAMchannels, power distribution among different OAM modes is analyzedthrough a modal decomposition approach, which corresponds to the casewhere the received OAM beams are demultiplexed without power loss andthe power of a desired OAM channel is completely collected by itsreceiver 3410.

Beam Divergence

For a communication link, it is generally preferable to collect as muchsignal power as possible at the receiver to ensure a reasonablesignal-to-noise ratio (SNR). Based on the diffraction theory, it isknown that a collimated OAM beam diverges while propagating in freespace. Given the same spot size of three cm at the transmitter, an OAMbeam with a higher azimuthal index diverges even faster, as shown inFIG. 35A. On the other hand, the receiving optical element usually has alimited aperture size and may not be able to collect all of the beampower. The calculated link power loss as a function of receiver aperturesize is shown in FIG. 35B, with different transmission distances andvarious transmitted beam sizes. Unsurprisingly, the power loss of a 1-kmlink is higher than that of a 100-m link under the same transmitted beamsize due to larger beam divergence. It is interesting to note that asystem with a transmitted beam size of 3 cm suffers less power loss thanthat of 1 cm and 10 cm over a 100-m link. The 1-cm transmitted beamdiverges faster than the 3 cm beam due to its larger diffraction.However, when the transmitted beam size is 10 cm, the geometricalcharacteristics of the beam dominate over the diffraction, thus leadinglarger spot size at the receiver than the 3 cm transmitted beam. Atrade-off between the diffraction, geometrical characteristics and thenumber of OAMs of the beam therefore needs to be carefully considered inorder to achieve a proper-size received beam when designing a link.

Misalignment Tolerance

Referring now to FIGS. 36A-36C, besides the power loss due tolimited-size aperture and beam divergence, another issue that needsfurther discussion is the potential misalignment between the transmitterand the receiver. In an ideal OAM multiplexed communication link,transmitter and receiver would be perfectly aligned, (i.e., the centerof the receiver would overlap with the center of the transmitted beam3602, and the receiver plane 3604 would be perpendicular to the lineconnecting their centers, as shown in FIG. 36A). However, due todifficulties in aligning because of substrate distances, and jitter andvibration of the transmitter/receiver platform, the transmitter andreceiver may have relative lateral shift (i.e., lateral displacement)(FIG. 36B) or angular shift (i.e., receiver angular error) (FIG. 36C).Both types of misalignment may lead to degradation of systemperformance.

Focusing on a link distance of 100 m, FIGS. 37A and 37B show the powerdistribution among different OAM modes due to lateral displacement andreceiver angular error when only l=+3 is transmitted with a transmittedbeam size of 3 cm. In order to investigate the effect of misalignment,the receiver aperture size is chosen to be 10 cm, which could cover thewhole OAM beam at the receiver. As the lateral displacement or receiverangular error increases, power leakage to other modes (i.e., channelcrosstalk) increases while the power on l=+3 state decreases. This isbecause larger lateral displacement or receiver angular causes largerphase profile mismatch between the received OAM beams and receiver. Thepower leakage to l=+1 and l=+5 is greater than that of l=+2 and l=+3 dueto their larger mode spacing with respect to l=+3. Therefore, a systemwith larger mode spacing (which also uses higher order OAM statessuffers less crosstalk. However, such a system may also have a largerpower loss due to the fast divergence of higher order OAM beams, asdiscussed above. Clearly, this trade-off between channel crosstalk andpower loss shall be considered when choosing the mode spacing in aspecific OAM multiplexed link.

An additional configuration in which the optical angular momentumprocessing and multi-layer overlay modulation technique described hereinabove may prove useful within the optical network framework is use withfree-space optics communications. Free-space optics systems provide anumber of advantages over traditional RF based systems from improvedisolation between the systems, the size and the cost of thereceivers/transmitters, need for an FCC license, and by combining space,lighting, and communication into the same system. Referring now to FIG.38, there is illustrated an example of the operation of a free-spacecommunication system. The free-space communication system utilizes afree-space optics transmitter 3802 that transmits a light beam 3804 to afree-space optics receiver 3806. The major difference between afiber-optic network and a free-space optic network is that theinformation beam is transmitted through free space rather than over afiber-optic cable. This causes a number of link difficulties, which willbe more fully discussed herein below. However, because the free spacesystem does not have the optic fiber to act as a waveguide, it is moresusceptible to the problems outlined above. Free-space optics is a lineof sight technology that uses the invisible beams of light to provideoptical bandwidth connections that can send and receive up to 2.5 Gbpsof data, voice, and video communications between a transmitter 3802 anda receiver 3806. Free-space optics uses the same concepts asfiber-optics, except without the use of a fiber-optic cable. Free-spaceoptics systems provide the light beam 3804 within the infrared (IR)spectrum, which is at the low end of the light spectrum. Specifically,the optical signal is in the range of 300 Gigahertz to 1 Terahertz interms of wavelength.

Presently existing free-space optics systems can provide data rates ofup to 10 Gigabits per second at a distance of up to 2.5 kilometers. Inouter space, the communications range of free space opticalcommunications is currently on the order of several thousand kilometers,but has the potential to bridge interplanetary distances of millions ofkilometers, using optical telescopes as beam expanders. In January of2013, NASA used lasers to beam an image of the Mona Lisa to the LunarReconnaissance Orbiter roughly 240,000 miles away. To compensate foratmospheric interference, an error correction code algorithm, similar tothat used within compact discs, was implemented.

Referring now to FIG. 39, there is illustrated a block diagram of afree-space optics system using orbital angular momentum and multileveloverlay modulation according to the present disclosure. The OAM twistedsignals, in addition to being transmitted over fiber, may also betransmitted using free optics. In this case, the transmission signalsare generated within transmission circuitry 3902 at each of the FSOtransceivers 3904. Free-space optics technology is based on theconnectivity between the FSO based optical wireless units, eachconsisting of an optical transceiver 3904 with a transmitter 3902 and areceiver 3906 to provide full duplex open pair and bidirectional closedpairing capability. Each optical wireless transceiver unit 3904additionally includes an optical source 3908 plus a lens or telescope3910 for transmitting light through the atmosphere to another lens 3910receiving the information. At this point, the receiving lens ortelescope 3910 connects to a high sensitivity receiver 3906 via opticalfiber 3912. The transmitting transceiver 3904 a and the receivingtransceiver 3904 b have to have line of sight to each other and bealigned both laterally and angularly. Obstacles, such as, trees,buildings, animals, and atmospheric conditions, all can hinder the lineof sight needed for this communications medium. Since line of sight isso critical, some systems make use of beam divergence or a diffused beamapproach, which involves a large field of view that toleratessubstantial line of sight interference without significant impact onoverall signal quality. The system may also be equipped with autotracking mechanism 3914 that maintains a tightly focused beam on thereceiving transceiver 3404 b, even when the transceivers are mounted ontall buildings or other structures that sway.

The modulated light source used with optical source 3908 is typically alaser or light emitting diode (LED) providing the transmitted opticalsignal that determines all the transmitter capabilities of the system.Only the detector sensitivity within the receiver 3906 plays an equallyimportant role in total system performance. For telecommunicationspurposes, only lasers that are capable of being modulated at 20 Megabitsper second to 2.5 Gigabits per second can meet current marketplacedemands. Additionally, how the device is modulated and how muchmodulated power is produced are both important to the selection of thedevice. Lasers in the 780-850 nm and 1520-1600 nm spectral bands meetfrequency requirements.

Commercially available FSO systems operate in the near IR wavelengthrange between 750 and 1600 nm, with one or two systems being developedto operate at the IR wavelength of 10,000 nm. The physics andtransmissions properties of optical energy as it travels through theatmosphere are similar throughout the visible and near IR wavelengthrange, but several factors that influence which wavelengths are chosenfor a particular system.

The atmosphere is considered to be highly transparent in the visible andnear IR wavelength. However, certain wavelengths or wavelength bands canexperience severe absorption. In the near IR wavelength, absorptionoccurs primarily in response to water particles (i.e., moisture) whichare an inherent part of the atmosphere, even under clear weatherconditions. There are several transmission windows that are nearlytransparent (i.e., have an attenuation of less than 0.2 dB perkilometer) within the 700-10,000 nm wavelength range. These wavelengthsare located around specific center wavelengths, with the majority offree-space optics systems designed to operate in the windows of 780-850nm and 1520-1600 nm.

Wavelengths in the 780-850 nm range are suitable for free-space opticsoperation and higher power laser sources may operate in this range. At780 nm, inexpensive CD lasers may be used, but the average lifespan ofthese lasers can be an issue. These issues may be addressed by runningthe lasers at a fraction of their maximum rated output power which willgreatly increase their lifespan. At around 850 nm, the optical source3908 may comprise an inexpensive, high performance transmitter anddetector components that are readily available and commonly used innetwork transmission equipment. Highly sensitive silicon (SI) avalanchephotodiodes (APD) detector technology and advanced vertical cavityemitting laser may be utilized within the optical source 3908.

VCSEL technology may be used for operation in the 780 to 850 nm range.Possible disadvantage of this technology include beam detection throughthe use of a night vision scope, although it is still not possible todemodulate a perceived light beam using this technique.

Wavelengths in the 1520-1600 nm range are well-suited for free-spacetransmission, and high quality transmitter and detector components arereadily available for use within the optical source block 3908. Thecombination of low attenuation and high component availability withinthis wavelength range makes the development of wavelength divisionmultiplexing (WDM) free-space optics systems feasible. However,components are generally more expensive and detectors are typically lesssensitive and have a smaller receive surface area when compared withsilicon avalanche photodiode detectors that operator at the 850 nmwavelength. These wavelengths are compatible with erbium-doped fiberamplifier technology, which is important for high power (greater than500 milliwatt) and high data rate (greater than 2.5 Gigabytes persecond) systems. Fifty to 65 times as much power can be transmitted atthe 1520-1600 nm wavelength than can be transmitted at the 780-850 nmwavelength for the same eye safety classification. Disadvantages ofthese wavelengths include the inability to detect a beam with a nightvision scope. The night vision scope is one technique that may be usedfor aligning the beam through the alignment circuitry 3914. Class 1lasers are safe under reasonably foreseeable operating conditionsincluding the use of optical instruments for intrabeam viewing. Class 1systems can be installed at any location without restriction.

Another potential optical source 3908 comprised Class 1M lasers. Class1M laser systems operate in the wavelength range from 302.5 to 4000 nm,which is safe under reasonably foreseeable conditions, but may behazardous if the user employs optical instruments within some portion ofthe beam path. As a result, Class 1M systems should only be installed inlocations where the unsafe use of optical aids can be prevented.Examples of various characteristics of both Class 1 and Class 1M lasersthat may be used for the optical source 3908 are illustrated in Table 2below.

TABLE 2 Laser Power Aperture Size Distance Power Density Classification(mW) (mm) (m) (mW/cm²) 850-nm Wavelength Class 1 0.78 7 14 2.03 50 20000.04 Class 1M 0.78 7 100 2.03 500 7 14 1299.88 50 2000 25.48 1550-nmWavelength Class 1 10 7 14 26.00 25 2000 2.04 Class 1M 10 3.5 100 103.99500 7 14 1299.88 25 2000 101.91

The 10,000 nm wavelength is relatively new to the commercial free spaceoptic arena and is being developed because of better fog transmissioncapabilities. There is presently considerable debate regarding thesecharacteristics because they are heavily dependent upon fog type andduration. Few components are available at the 10,000 nm wavelength, asit is normally not used within telecommunications equipment.Additionally, 10,000 nm energy does not penetrate glass, so it isill-suited to behind window deployment.

Within these wavelength windows, FSO systems should have the followingcharacteristics. The system should have the ability to operate at higherpower levels, which is important for longer distance FSO systemtransmissions. The system should have the ability to provide high speedmodulation, which is important for high speed FSO systems. The systemshould provide a small footprint and low power consumption, which isimportant for overall system design and maintenance. The system shouldhave the ability to operate over a wide temperature range without majorperformance degradations such that the systems may prove useful foroutdoor systems. Additionally, the mean time between failures shouldexceed 10 years. Presently existing FSO systems generally use VCSELS foroperation in the shorter IR wavelength range, and Fabry-Perot ordistributed feedback lasers for operation in the longer IR wavelengthrange. Several other laser types are suitable for high performance FSOsystems.

A free-space optics system using orbital angular momentum processing andmulti-layer overlay modulation would provide a number of advantages. Thesystem would be very convenient. Free-space optics provides a wirelesssolution to a last-mile connection, or a connection between twobuildings. There is no necessity to dig or bury fiber cable. Free-spaceoptics also requires no RF license. The system is upgradable and itsopen interfaces support equipment from a variety of vendors. The systemcan be deployed behind windows, eliminating the need for costly rooftopsites. Further, it is easier to deploy in buildings as the system can belocated as the area requires, saving significant costs of running cablesto rooftops. It is also immune to radiofrequency interference orsaturation. The system is also fairly speedy. The system provides 10Gigabits per second of data throughput. This provides ample bandwidth totransfer files between two sites. With the growth in the size of files,free-space optics provides the necessary bandwidth to transfer thesefiles efficiently.

Free-space optics also provides a secure wireless solution. The laserbeam cannot be detected with a spectral analyzer or RF meter. The beamis invisible, which makes it difficult to find. The laser beam that isused to transmit and receive the data is very narrow. This means that itis almost impossible to intercept the data being transmitted. One wouldhave to be within the line of sight between the receiver and thetransmitter in order to be able to accomplish this feat. If this occurs,this would alert the receiving site that a connection has been lost orthe amount of signal received severely diminished. Thus, minimalsecurity upgrades would be required for a free-space optics system.

However, there are several weaknesses with free-space optics systems.The distance of a free-space optics system is very limited. Currentlyoperating distances are approximately within 2 kilometers. Although thisis a powerful system with great throughput, the limitation of distanceis a big deterrent for full-scale implementation. Further, the more OAMsapplied, the greater divergence over distance. Additionally, all systemsrequire line of sight be maintained at all times during transmission.Any obstacle, be it environmental or animals can hinder thetransmission. Free-space optic technology must be designed to combatchanges in the atmosphere which can affect free-space optic systemperformance capacity. Finally, any shift in the mounting apparatus cancause the beam to be misaligned. Shifts can be caused by wind,earthquakes, ground shifting and even traffic.

Referring now to FIGS. 40A through 40D, in order to achieve higher datacapacity within optical links, an additional degree of freedom frommultiplexing multiple data channels must be exploited. Moreover, theability to use two different orthogonal multiplexing techniques togetherhas the potential to dramatically enhance system performance andincreased bandwidth.

One multiplexing technique which may exploit the possibilities is modedivision multiplexing (MDM) using orbital angular momentum (OAM). OAMmode refers to laser beams within a free-space optical system orfiber-optic system that have a phase term of e^(ilφ) in their wavefronts, in which (p is the azimuth angle and l determines the OAM value(topological charge). In general, OAM modes have a “donut-like” ringshaped intensity distribution. Multiple spatial collocated laser beams,which carry different OAM values, are orthogonal to each other and canbe used to transmit multiple independent data channels on the samewavelength. Consequently, the system capacity and spectral efficiency interms of bits/S/Hz can be dramatically increased. Free-spacecommunications links using OAM may support 100. Tbits/capacity. Varioustechniques for implementing this as illustrated in FIGS. 40A through 40Dinclude a combination of multiple beams 4002 having multiple differentOAM values 4004 on each wavelength. Thus, beam 4002 includes OAM values,OAM1 and OAM4. Beam 4006 includes OAM value 2 and OAM value 5. Finally,beam 4008 includes OAM3 value and OAM6 value. Referring now to FIG. 40B,there is illustrated a single beam wavelength 4010 using a first groupof OAM values 4012 having both a positive OAM value 4012 and a negativeOAM value 4014. Similarly, OAM2 value may have a positive value 4016 anda negative value 4018 on the same wavelength 4010. While mode divisionmultiplexing of OAM modes is described above, other orthogonal functionsmay be used with mode division multiplexing such as Laguerre Gaussianfunctions, Hermite Gaussian functions, Jacobi functions, Gegenbauerfunctions, Legendre functions and Chebyshev functions.

FIG. 40C illustrates the use of a wavelength 4020 having polarizationmultiplexing of OAM value. The wavelength 4020 can have multiple OAMvalues 4022 multiplexed thereon. The number of available channels can befurther increased by applying left or right handed polarization to theOAM values. Finally, FIG. 40D illustrates two groups of concentric rings4060, 4062 for a wavelength having multiple OAM values.

Another multiplexing technique is wavelength distribution multiplexing(WDM), WDM has been widely used to improve the optical communicationcapacity within both fiber-optic systems and free-space communicationsystem. Combining OAM and WDM has not previously been done. However, OAMmode multiplexing and WDM are mutually orthogonal such that they can becombined to achieve a dramatic increase in system capacity. Referringnow to FIG. 41, there is illustrated a scenario where each WDM channel4102 contains many orthogonal OAM beam 4104. Thus, using a combinationof orbital angular momentum with wave division multiplexing, asignificant enhancement in communication link to capacity may beachieved.

Current optical communication architectures have considerable routingchallenges. A routing protocol for use with free-space optic system musttake into account the line of sight requirements for opticalcommunications within a free-space optics system. However, an opticsnetwork may be modeled as a directed hierarchical random sectorgeometric graph in which sensors route their data via multi-hop paths toa base station through a cluster head. This technique is a new efficientrouting algorithm for local neighborhood discovery and a base stationuplink and downlink discovery algorithm. The routing protocol requiresorder O log(n) storage at each node versus order O(n) used withincurrent techniques and architectures. This new technique has theadvantage of being much faster than current systems.

Current routing protocols are based on link state, distance vectors,path vectors, or source routing, and they differ from the new routingtechnique in significant manners. First, current techniques assume thata fraction of the links are bidirectional. This is not true within afree-space optic network in which links are unidirectional. Second, manycurrent protocols are designed for ad hoc networks in which the routingprotocol is designed to support multi-hop communications between anypair of nodes. The goal of the sensor network is to route sensorreadings to the base station. Therefore, the dominant traffic patternsare different from those in an ad hoc network. In a sensor network, nodeto base stations, base station to nodes, and local neighborhoodcommunication are mostly used.

Many paths of wireless and free space network are unidirectional. Recentstudies on wireless and free space optical systems show that as many as5 percent to 10 percent of links and wireless ad hoc networks areunidirectional due to various factors. Routing protocols such as DSDVand AODV use a reverse path technique, implicitly ignoring suchunidirectional links and are therefore not relevant in this scenario.Other protocols such as DSR, ZRP, or ZRL have been designed or modifiedto accommodate unidirectionality by detecting unidirectional links andthen providing bidirectional abstraction for such links.Unidirectionality only allows information transmission in a singledirection which does not enable a response to be provided to aninformation transmission system. Referring now to FIG. 42, one solutionfor dealing with unidirectionality is tunneling, in whichbidirectionality is emulated for a unidirectional link by usingbidirectional links on a reverse back channel to establish the tunnel.Tunneling also prevents implosion of acknowledgement packets and loopingby simply pressing link layer acknowledgements for tunneled packetsreceived on a unidirectional link. Tunneling, however, works well inmostly bidirectional networks with few unidirectional links.

Within a network using only unidirectional links such as a free-spaceoptical network, systems such as that illustrated in FIGS. 42 and 43would be more applicable. Nodes within a unidirectional network utilizea directional transmit 4202 transmitting from the node 4200 in a single,defined direction. Additionally, each node 4200 includes anomnidirectional receiver 4204 which can receive a signal coming to thenode in any direction. Also, as discussed here and above, the node 4200would also include a 0 log(n) storage 4206. Thus, each node 4200 provideonly unidirectional communications links. Thus, a series of nodes 4200as illustrated in FIG. 43 may unidirectionally communicate with anyother node 4200 and forward communication from one location to anotherthrough a sequence of interconnected nodes.

Multiplexing of the topological charge to the RF as well as free spaceoptics in real time provides redundancy and better capacity. Whenchannel impairments from atmospheric disturbances or scintillationimpact the information signals, it is possible to toggle between freespace optics to RF and back in real time. This approach still usestwisted waves on both the free space optics as well as the RF signal.Most of the channel impairments can be detected using a control or pilotchannel and be corrected using algorithmic techniques (closed loopcontrol system) or by toggling between the RF and free space optics.

Topological charge may be multiplexed to the wave length for eitherlinear or circular polarization. In the case of linear polarizations,topological charge would be multiplexed on vertical and horizontalpolarization. In case of circular polarization, topological charge wouldbe multiplexed on left hand and right hand circular polarizations.

The topological charges can be created using Spiral Phase Plates (SPPs)such as that illustrated in FIG. 11E, phase mask holograms or a SpatialLight Modulator (SLM) by adjusting the voltages on SLM which createsproperly varying index of refraction resulting in twisting of the beamwith a specific topological charge. Different topological charges can becreated and muxed together and de-muxed to separate charges.

As Spiral Phase plates can transform a plane wave (l=0) to a twistedwave of a specific helicity (i.e. l=+1), Quarter Wave Plates (QWP) cantransform a linear polarization (s=0) to circular polarization (i.e.s=+1).

Cross talk and multipath interference can be reduced usingMultiple-Input-Multiple-Output (MIMO).

Most of the channel impairments can be detected using a control or pilotchannel and be corrected using algorithmic techniques (closed loopcontrol system).

In a further embodiment illustrated in FIG. 44, both RF signals and freespace optics may be implemented within a dual RF and free space opticsmechanism 4402. The dual RF and free space optics mechanism 4402 includea free space optics projection portion 4404 that transmits a light wavehaving an orbital angular momentum applied thereto with multileveloverlay modulation and a RF portion 4406 including circuitry necessaryfor transmitting information with orbital angular momentum andmultilayer overlay on an RF signal 4410. The dual RF and free spaceoptics mechanism 4402 may be multiplexed in real time between the freespace optics signal 4408 and the RF signal 4410 depending upon operatingconditions. In some situations, the free space optics signal 4408 wouldbe most appropriate for transmitting the data. In other situations, thefree space optics signal 4408 would not be available and the RF signal4410 would be most appropriate for transmitting data. The dual RF andfree space optics mechanism 4402 may multiplex in real time betweenthese two signals based upon the available operating conditions.

Multiplexing of the topological charge to the RF as well as free spaceoptics in real time provides redundancy and better capacity. Whenchannel impairments from atmospheric disturbances or scintillationimpact the information signals, it is possible to toggle between freespace optics to RF and back in real time. This approach still usestwisted waves on both the free space optics as well as the RF signal.Most of the channel impairments can be detected using a control or pilotchannel and be corrected using algorithmic techniques (closed loopcontrol system) or by toggling between the RF and free space optics.

Focusing OAM Signals

When applying higher order orbital angular momentum (OAM) values tooptical or RF beams, divergence issues can cause problems betweentransmitting and receiving units as shown in FIGS. 35A and 35B. Asillustrated in FIG. 45, there are illustrated transmissions between atransmitter 4502 and a receiver 4504 of an RF or optical beam 4506. Forlower order values of beam helicity, for example, l=1, the beam 4506A isfairly focused and the entire beam can be transmitted from thetransmitter 4502 to the receiver 4508. However, as the helicity of thebeam increases for l=2 up to f=n, as illustrated by beams 4506B and4506C, the divergence of the beam 4506 increases enabling less and lessof the beam transmitted from the transmitter 4502 to be received by thereceiver 4504. When a beam becomes divergent to such a level that mostof the beam 4506C may not be received by the receiver 4504 a great dealof data may be lost as only a portion of the transmission beam 4506C maybe received. Thus, in order to extend the transmission distance andcapacity of a communication or radar system using free-space optics orRF communications using orbital angular momentum processing, there is aneed to more narrowly focus the OAM based multiplexed beam 4506 sincethe beams are diverging for higher values of beam helicity.

One manner for more particularly focusing higher value helicity RF beamsutilizes an antenna array is illustrated in FIG. 46 for focusing thehigher value helicity OAM processed beam 4602. The system such as thatillustrated in FIG. 46 overcomes the problem of beam divergence usingfocused OAM beams as well as the application of non-sinusoidalultra-wide band techniques. The development of ultra-wide band largecurrent radiator (LCR) antennas has made it possible to radiatenanosecond wide impulses within inexpensive CMOS chips. As discussedpreviously, the generation of an OAM processed beam involves the receiptof multiple data streams 4604. Each of the data streams 4604 areprocessed by modulation/demodulation circuitry 4606 to provide amodulation scheme to the data stream, for example, the multiple leveloverlay modulation technique described herein above. Themodulator/demodulator circuitries 4606 provide the modulated signals tothe OAM signal processing circuitry 4608 that applies the OAM twist tothe data streams and combines them into a single output OAM processeddata stream 4610. The multiplex OAM signal 4610 is provided to an RF oroptical transmission array 4612 controlled in accordance with controlsignals from an array controller 4614 such that the OAM multiplex beam4610 may be more particularly focused into the focus beam 4602 as willbe more fully described herein below.

By providing a more focused OAM processed RF or optical beam, a numberof applications may become more practical for implementation using theOAM focused beam. The first of these include an OAM processed optical orRF beam used in a ground penetrating application such as thatillustrated in FIG. 47. In this application, the ground based OAMfocused system 4702 produces a focused OAM beam 4704 that is radiatedinto the ground 4706 for detecting below-ground materials andstructures. The ground penetrating system can be reengineered andrepackaged for a number of applications. Examples of these applicationsinclude the energy industry for use in oil and gas exploration. Theground penetrating radar can be used within the water industry for waterexploration. Other typical applications are the locations ofnon-metallic pipes in the ground, cavities under roads and railroad bedsand location of reinforcement iron bars in concrete walls of atomicpower stations. Military applications of the ground penetratingapplication would include detection of hidden arms piles, land mines,unexploded ordinance and underground tunnels or detection of undergroundexplosive devices. Archeological explorations would also find usefulapplications of the ground penetrating application. The groundpenetrating application could also be used within ship navigation forthe detection of ice thicknesses for ship navigation in order to findthinner ice for passing of ice breakers or in the determination ofice-thickness for ground based transportation over ice that is thickenough to support a load. Geophysical applications would include theexploration of earth layers for geophysical analysis. While the abovelist provides only a partial example of a number of applications for usewith a ground penetrating radar, any other number of possibleapplications would be understood by one skilled in the art.

Ground probing radars have been developed since the 1960's. A commonfeature of ground probing radars is that they do not use a sinuosoidalcarrier, in order to reduce attenuation by absorption within the ground.Ground penetrating radars penetrate soil, rock, sand, ice and freshwater to depths of a few meters. A probing depth of 10 meters is anupper limit under exceptional and optimal conditions usually involvingan absence of moisture.

The increase of penetration depth of ground probing radars has beenhandicapped by two things. First, there is no significant market forradars that reach depths of more than 10 meters until one reachesseveral hundred meters, and the radar becomes suitable for geologicalexploration. The energy required for a radar pulse to reach a distance rand produce a useful return signal increases linearly with the distancer due to absorption and “geometric spreading.” The absorption is not aserious problem. If a resolution of about Δr=10 cm is desired at adistance of r=10 meters or a ratio Δr/r=0.01, a pulse of a nominalduration of 1 nanosecond is needed. For r=1000 meters, and Δr/r=0.01, weget Δr=10 meters and a nominal pulse duration of 100 nanoseconds isobtained. Since a resolution of 10 meters at a distance of 1000 metersappears sufficient for geological probing, the increase in the requiredpulse energy by a factor of 100 can be achieved by making the pulse 100times longer. No increase of the peak power is required. The importantproblem is the geometric spreading of the wave. An increase of theprobing depth from 10 meters to 100 meters calls for 10⁴=10,000 timesthe energy to overcome geometric spreading, and a distance of 1,000meters requires 100⁴=100,000,000 times the energy.

In order to overcome the two effects blocking the development of deepground probing radars, two things must be accomplished. First, there isno point in developing ground probing radar incrementally for depths of20 meters, 40 meters, etc., but one must reach several hundred metersbefore there is a market for the radar. Additionally, the decrease ofthe received energy due to the r⁴ or 1/r⁴ law can be reduced to 1/r² dueto the principle of the “focused radar.” Such a radar will give goodimages of what there is at certain depths, for example, 500 meters, butobscure images from lesser or greater depths. The principle works forstationary objects or targets that are detectable by a ground probingradar, but is if no use for surveillance radars that are used to detectmoving objects. The attenuation loss due to geometric spreadingproportionate to 1/r² is 100 for an increase of distance from 10 to 100meters and 10,000 for an increase to 1,000 meters. Thus, the requiredpower becomes on the order of one kilowatt. Such a value is withintechnical capabilities of existing technologies.

Another useful application of the focused OAM beam is with backhaul anddata sensor applications. Referring now to FIG. 48, there is illustrateda microwave/free space system 4802 providing an output unfocused OAMbeam to an OAM beam focusing system 4804. The OAM beam focusing system4804 provides a single point to point broadcast to a fixed receiver 4806for example in a microwave backhaul system. Additionally, as shown inFIG. 49, a microwave/free space optical system 4902 may provide anunfocused OAM processed beam to a OAM beam focusing system 4904 whichmay provide a point to multipoint broadcast to multiple receivers 4906within a free space optical communications or microwave backhaul system.

Referring now to FIG. 50, there is illustrated a block diagram of anantenna array 5002 that may be used for providing a focused OAMprocessed beam in accordance with the present disclosure. The antennaarray 5002 receives a data stream including multiplexed OAM processeddata 5004. The OAM processed data 5004 is applied to each antenna 5006within the antenna array 5002. Each of the antennas 5006 transmit theOAM processed data 5004 in accordance with transmissions instructionsprovided from the array transmission control block 5008. The arraytransmission control block 5008 controls transmission of the data in adefined fashion in order to control the transmissions to provide a morefocused beam that overcomes the divergence inherent in higher order OAMprocessed signals. The particular manner for controlling thetransmission of the OAM processed data will be more fully describedherein below.

The manner for focusing an OAM processed beam may be illustrated bymeans of ray optics. Referring now to FIG. 51, there is illustrated apoint-like light source 5102 denoted radiator R at location y_(R) in thefocal plane f₁ 5104 of a lens 5106. The focal plane 5104 is a distancex_(f) from the center x=0 of the lens 5106. A light beam 5108 is formedthat has an angle β 5110 relative to the lens axis 5112. The value of βis given by the formula:

$\begin{matrix}{{\tan \; \beta} = \frac{y_{R}}{x_{f}}} & (14)\end{matrix}$

In terms of ray optics, the power or energy density of the radiationpropagating through a plane perpendicular to the radiation axis does notdepend on the distance r from the radiator. The transition from rayoptics to wave optics brings a decrease of the power or energy densityproportionate to 1/r². For sinusoidal waves, one must use the concept ofenergy density since a periodic, infinitely extended sine wave hasinfinite energy. Waves representing signals which include sinusoidalpulses with a finite number of cycles have a finite energy and energydensity, their beam patterns differ from those of sinusoidal waves, buttheir energy density decreases like 1/r² just like the power density ofsinusoidal waves.

Referring now to FIG. 52, in this image the Radiator® 5202 is notlocated in the focal plane 5204 of the lens 5206 but further away fromthe focal plane 5204 and the radiator plane 5208. An image (I) 5210 ofthe radiator 5202 is produced in the image plane 5212. Thus, all of theradiation is concentrated in one point at a distance x₁ from the lens5206 which is quite different from the divergent beam illustrated inFIG. 51. The angles β and γ in FIG. 51 are defined by the relationships:

$\begin{matrix}{{\tan \; \beta} = {\frac{y_{R}}{x_{R}} = \frac{y_{I}}{x_{I}}}} & (15) \\{{\tan \; \gamma} = {\frac{y_{R}}{x_{f}} = \frac{y_{I}}{x_{I} + x_{f}}}} & (16)\end{matrix}$

from which we get the x and y-coordinates of the image I 3710 of theradiator R 3702:

$\begin{matrix}{x_{I} = {{- \frac{x_{f}}{x_{R} - x_{f}}}x_{R}}} & (17) \\{y_{I} = {{- \frac{x_{f}}{x_{R} - x_{f}}}y_{R}}} & (18)\end{matrix}$

For x_(R)→x_(f) we get x_(I)→−∞ and y_(I)→−∞, which represents FIG. 36.The image of R 3702 is produced at an infinite distance x and isinfinitely far from the lens with sign reversal of the location y_(R) ofR.

When the radiator R 5202 of FIG. 52 is moved to the location y=0 in theradiator plane (see radiator R₀ 5214), the new radiator 5214 is denotedR₀ and its image is I₀ 5216. If the radiator R₀ 5214 is moved in theinterval 2x_(f)≥x_(R)>x_(f), the image I₀ 5216 will move in the interval−2x_(f)≥x_(I)>→∞. The velocity dx₁/dt of the image I₀ 5216 as a functionof the velocity dx_(R)/dt of the radiator R₀ 5214 is obtained by thedifferentiation of Equation 17:

$\begin{matrix}{\frac{{dx}_{I}}{dt} = {\frac{x_{f}^{2}}{\left( {x_{R} - x_{f}} \right)^{2}}\frac{{dx}_{R}}{dt}}} & (19)\end{matrix}$

At x_(R)=2x_(f), the image I₀ 5216 moves with the velocity of theradiator. While for x_(R)→x_(f), the velocity of the image I₀ 5216approaches ∞ since ray optics is not a relativistic theory. Radiator5214 and image 5216 always move in the same direction.

A comparison of FIGS. 51 and 52 suggest that variable focusing shouldprovide a narrower beam or angular resolution than fixed focusing atinfinity, provided the distances of interest are not too large comparedwith the diameter of the lens 5206. To investigate this possibilityfurther, the lens 5206 and radiator 5202 are replaced with an array ofradiators that emit electromagnetic waves with carefully chosen timevariation and timing as illustrated in FIG. 53. FIG. 53 illustrates aradiator array 5302 consisting of a plurality of radiating antennas 5304that radiate energy as described herein. A line of array antennas 5304includes 2 m=8 radiators that are a distance (|n|−½)d, n=±1, ±2, ±3, ±4apart from the array axis radiates electromagnetic waves whose electricand magnetic field strengths have the time variation of rectangularpulses 5306. These pulses 5306 are delayed in time with respect to eachother and should arrive at the image point I 5308 on the array axis 5310at a distance L 5312 from the center of the array at the same time. Thedistance x_(n) between the radiator 5302 and the image point I 5308 isgiven by the equation:

x _(n)=[L ²+(|n|−½)² d ²]^(1/2)  (20)

and the propagation time is x_(n)/c. In order to have the leading edgeof all pulses 5306 arrive simultaneously at the image point I 5308 atthe time t₀=L/c one must radiate the pulses 5306 from the radiator n5302 at the time

$\begin{matrix}{t_{n} = \frac{L - \left\lbrack {L^{2} + {\left( {{n} - \frac{1}{2}} \right)^{2}d^{2}}} \right\rbrack^{\frac{1}{2}}}{c}} & (21) \\{\approx {{- \frac{\left( {{2{n}} - 1} \right)^{2}}{8}}\frac{d^{2}}{Lc}\mspace{14mu} {for}\mspace{14mu} \left( {{2{n}} - 1} \right)^{2}{d^{2}/4}L^{2}{\operatorname{<<}1}}} & (22)\end{matrix}$

The time difference between the pulse radiation from radiator n 5302 andfrom radiator 1 is of practical interest:

$\begin{matrix}{{\Delta \; t_{n}} = {{t_{n} - t_{1}} = {\frac{L}{c}\left\{ {\left( {1 + \frac{d^{2}}{4L^{2}}} \right)^{\frac{1}{2}} - \left\lbrack {1 + {\left( {{n} - \frac{1}{2}} \right)^{2}\frac{d^{2}}{L^{2}}}} \right\rbrack^{\frac{1}{2}}} \right\}}}} & \; \\{\approx {{- \frac{\left( {{2{n}} - 1} \right)^{2}}{8}}\frac{d^{2}}{Lc}}} & (23) \\\begin{matrix}{{\Delta \; t_{n}} = {t_{n} - t_{1}}} \\{= {\frac{L}{c}\left\{ {\left( {1 + \frac{d^{2}}{4L^{2}}} \right)^{1/2} - \left\lbrack {1 + {\left( {{n} - \frac{1}{2}} \right)^{2}\frac{d^{2}}{L^{2}}}} \right\rbrack^{1/2}} \right\}}} \\{\approx {{- \frac{\left( {{2{n}} - 1} \right)^{2}}{8}}\frac{d^{2}}{Lc}}}\end{matrix} & (24)\end{matrix}$

If one wants a concentration of energy by focusing not in one imagepoint at the distance L 5312 but generally at k points L, L+ΔL, . . . ,L+(i−1)ΔL, . . . for i=1, 2, . . . , k one must replace the k=1 set of 2m pulses in FIG. 53 by k sets of 2 m pulses each. An example is given inFIG. 39 as indicated generally at 3902 for k=3. If the image point i3908 is produced at the distance L+ΔL at the time t₁=(L+ΔL)/c+ΔT, andgenerally at the location L+(i−1) ΔL at the time t_(i-1)=[L+(i−1)ΔL]/c+(i−1) ΔT. The extra time (i−1)ΔT is required to permit aselectable minimum time difference between the pulses of the radiators5304 n=±1. The generalization of the time t_(n) in Equation 23 tot_(n,1) becomes:

$\begin{matrix}{t_{n,i} = {{\frac{1}{c}\left\{ {L + {\left( {i - 1} \right)\Delta \; L} - \left\lbrack {\left\lbrack {L + {\left( {i - 1} \right)\Delta \; L}} \right\rbrack^{2} + {\left( {{n} - \frac{1}{2}} \right)^{2}d^{2}}} \right\rbrack^{1/2}} \right\}} + {\left( {i - 1} \right)\Delta \; T}}} & (25)\end{matrix}$

The generalization of Equation 9 does not contain ΔT:

$\begin{matrix}\begin{matrix}{{\Delta \; t_{n,i}} = {t_{n,i} - t_{1,i}}} \\{= \frac{L + {\left( {i - 1} \right)\Delta \; L}}{c}} \\{\begin{Bmatrix}{\left( {1 + \frac{d^{2}}{{4\left\lbrack {L + {\left( {i - 1} \right)\Delta \; L}} \right\rbrack}^{2}}} \right)^{\frac{1}{2}} -} \\\left\lbrack {1 + {\left( {{n} - \frac{1}{2}} \right)^{2}\frac{d^{2}}{\left\lbrack {L + {\left( {i - 1} \right)\Delta \; L}} \right\rbrack^{2}}}} \right\rbrack^{\frac{1}{2}}\end{Bmatrix}}\end{matrix} & (26)\end{matrix}$

In order to obtain some numerical values to judge the potentialapplications of focused waves we take from FIG. 53 the relation

d=D/2m  (27)

and substituted into Equation 24:

Δt _(m) =t _(m) −t ₁ ≈−D ²/8Lc  (28)

TABLE 3 CHARACTERISTIC TIME DELAYS ΔT_(M) REQUIRED TO PRODUCE A FOCUSEDWAVE AT THE DISTANCE L WITH A LINE ARRAY OF LENGTH D D L −Δt_(m) =D²/8Lc [m] [m] [ps] 1 10 41.7 100 4.17 1000 0.417 2 20 83.3 200 8.332000 0.833 5 50 208.3 500 20.8 5000 2.8 10 100 417 1000 41.7 10000 4.1720 200 833.3 2000 83.3 20000 8.33 50 500 2083 5000 208 50000 20.8 1001000 4167 10000 417 100000 41.7

Table 3 shows Δt_(m) as a function of D and L. A few values of Δt_(m)are larger than 1 ns and a few are smaller than 1 ps, but most lie inthe range from 1 ps to 1 ns. This determines the typical requiredaccuracy of the beginning of the pulses in FIGS. 53 and 54.

If one wants to produce one focused point I₀ in FIG. 53 one needs oneset of pulses (i=1, 2 . . . ) according to FIG. 54 at 5402 which shallhave the combined energy W. For k focused points at k differentdistances k sets of pulses are needed when combined with a combinedenergy kW. Hence, in principle the required energy increases linearlywith distance. The usual beamforming requires that the radiated energyincreases with the square of the distance if a certain energy density atthe chosen distance is to be achieved. For a numerical comparison of thetwo methods reference is now made to FIG. 55. An array R 5502 ofradiators radiates the energy W₀ into a spherical angle ε 5504 atdistance L 5506. The energy density ω_(L) is obtained according to theequation:

ω_(L) =W ₀ /∈L ²  (29)

while at the distance L+(k−1) ΔL we get the energy density ω_(L,k) isgenerated according to the equation:

ω_(L,k) =W ₀/∈[L+(k−1)ΔL]²  (30)

If the energy density ω_(L,k) is to be increased to the value of ω_(L)one must multiply W₀ by a factor of K:

K=[L+(k−1)ΔL]² /L ²=[1+(k−1)ΔL/L]²  (31)

If a focused wave makes the energy W flow through the point at distanceL 5506 in FIG. 55 the energy kW makes the energy W flow through each oneof the k points L, L+ΔL, . . . , L+(k−1)ΔL. If the condition

kW<LW ₀=[1+(k−1)ΔL/L]² W ₀  (32)

is satisfied the focused wave will require less energy than thespherical wave. Equation 17 may then be rewritten as:

$\begin{matrix}{0 < {k^{2} - {\left\lbrack {{\frac{W}{W_{0}}\left( \frac{L}{\Delta \; L} \right)^{2}} - {2\; \frac{L}{\Delta \; L}} + 2} \right\rbrack k} + {\left( \frac{\Delta \; L}{L} \right)^{2}\left( {1 - \frac{\Delta \; L}{L}} \right)^{2}}}} & (33)\end{matrix}$

If the sign < is replaced by an equality sign we obtain for the largerroot k₁ of k in the following second order approximation:

$\begin{matrix}{k_{1} \approx {{\frac{W}{W_{0}}\left( \frac{L}{\Delta \; L} \right)^{2}} - {2\; \frac{L}{\Delta \; L}}}} & (34)\end{matrix}$

As a first example, consider the values W=W₀, L/ΔL=10. From Equation 34we obtain the approximation k₁≈100−20=80. From the exact formula ofEquation 31, we derive the following values:

k→ 79 80 81 82 83 K→ 77.4 79.2 81 82.8 84.6

We see that k₁=81 rather than 80 is the exact value. For k>81 thefocused wave will require less radiated energy. The distance xcorresponds to k=81 follows from FIG. 40:

x=L+(k−1)ΔL=L(1+80×0.1)=9L  (35)

As a second example consider W=W₀, L/ΔL=100. Equation 34 yields k₁≈9800.The exact formula 16 yields:

k→ 9800 9801 9802 K→ 9799.0 9801 9802.9

The exact value of k₁ is 9801 rather than 9800. The focused waverequires less energy for k>9801 and the distance x corresponds to k=9801becomes:

x=L+(k−1)ΔL=L(1+9800×0.01)=99L  (36)

Comparison with Equation 36 shows that the distance has increased by afactor of 11 while the focus point increased by a factor of 9800/81=121.

One of the best features of focused waves applied to radar is that theenergy of a pulse returned by a reflector does not vary with thedistance r like 1/r² as would be the case for an unfocused wave.Similarly, the energy of a pulse returned by a point-like scattererdecreases like 1/r² rather than 1/r⁴. The principle of this effect canreadily be shown by means of ray optics. Consider FIG. 56 at 5602 whichshows a radiator R 5604, a lens 5606, an image point I_(R) 5608 and areflector 5610 at a distance r between the lens 5606 and the image point5608. A perfect reflector 5610 perpendicular to the optical axis 5612will return all the energy coming from reflector R 5610 to the lens 5606to the new image point I′_(R) 5614 and via the lens to R′ 5616. If thereflector 5610 is located at the image point I_(R) 5608 as illustratedat FIG. 56B the points I_(R) 5608 and I′_(R) 5614 as well as point R5604 and point R′ 5616 will coincide.

FIG. 56C illustrates the case of the reflector 5610 having a largerdistance r from the lens 5606 than the image point I_(R) 5608. Only partof the incident energy is returned and focused on the point R′ 5616. Thereturned energy decreases with the distance r proportional to 1/r². Theuse of a larger lens 5606 for the returned signal would avoid thedecrease.

If the reflector 5610 in FIG. 56 is replaced with a point-like scatterer5702 we have the configuration illustrated in FIG. 57. The returnedenergy decreases now like 1/r² since the scatterer 5702 produces aspherical wave 5704 whose surface increases like r². Using FIG. 56 onemay readily generalize this illustration to the case when the scatterer5702 is to the left or right of the image point of the radiation source.

FIG. 58 illustrates that focusing can greatly improve the angularresolution of a radar. An emitting radar 5802 assumes unfocused waves5804 for a ground probing radar that is mounted on a cart. The cart ispulled along the surface of the ground 5806. A large area is illuminatedat a distance D−ΔD≤d≤D+ΔD. Radar 5808 shows the same radar using focusedwaves 5810. The improved angular resolution in a layer at the depth ofD±ΔD is striking.

An electromagnetic wave travels along a stripline with a velocitybetween the velocity c of light and c/2. In 41.7 ps it travels 1.25 cmor less. An array of the dimensions 10 meters×10 meters built totolerances of less than 1 cm would be required to reach a depth of 1000meters according to Table 3. It would be difficult to build such anarray and even more difficult to use it in the field. The way aroundthis problem does not require a rigid array. We only have to know whereeach radiator and sensor is at any given time. According to Table 1, anarray of 100 m×100 m would need a basic timing accuracy of only 4.167ns. The location technology presently existing permits one to track thelocation of n radiators/sensors distributed over an area of 100 m×100 mto much better than 4 ns or 1.2 m. This permits one to time radiationand reception at the n radiators/sensors as if they were mounted on arigid array. Hence, the array sized up to 100 m×100 m listed in Table 3are perfectly realistic if one does not think in terms of a rigid arraybut an array with an electronically monitored location ofradiators/sensors.

Diversions within OAM beams may also be reduced using phased arrays. Byusing multiple transmitting elements in a geometrical configuration andcontrolling the current and phase for each transmitting element, theelectrical size of the antenna increases as does the performance of theantenna. The antenna system created by two or more individual intendedelements is called an antenna array. Each transmitting element does nothave to be identical but for simplification reasons the elements areoften alike. To determine the properties of the electric field from anarray the array factor (AF) is utilized.

The total field from an array can be calculated by a superposition ofthe fields from each element. However, with many elements this procedureis very unpractical and time consuming. By using different kinds ofsymmetries and identical elements within an array, a much simplerexpression for the total field may be determined. This is achieved bycalculating the so-called array factor (AF) which depends on thedisplacement (and shape of the array), phase, current amplitude andnumber of elements. After calculating the array factor, the total fieldis obtained by the pattern multiplication rule which is such that thetotal field is the product of the array factor in the field from onesingle element.

E _(total) =E _(single element) ×AF

This formula is valid for all arrays consisting of identical elements.The array factor does not depend on the type of elements used, so forcalculating AF it is preferred to use point sources instead of theactual antennas. After calculating the AF, the equation above is used toobtain the total field. Arrays can be 1D (linear), 2D (planar) or 3D. Ina linear array, the elements are placed along the line and in a planarthey are situated in a plane.

Referring now to FIG. 59, there is illustrated in the manner in whichHermite Gaussian beams and Laguerre Gaussian beams will diverge whentransmitted from a phased array of antennas. For the generation ofLaguerre Gaussian beams a circular symmetry over the cross-section ofthe phased antenna array is used, and thus, a circular grid will beutilized. For the generation of Hermite Gaussian beams 5902, arectangular array 5904 of array elements 5906 is utilized. As can beseen with respect to FIG. 59, the Hermite Gaussian waves 5908 provide amore focused beam front then the Laguerre Gaussian waves 5910.

Reduced beam divergence may also be accomplished using a pair of lenses.As illustrated in FIG. 60A, a Gaussian wave 6002 passing through aspiral phase plate 6004 generates an output Laguerre Gaussian wave 6006.The Laguerre Gaussian wave 6006 when passing from a transmitter aperture6008 to a receiver aperture 6010 diverges such that the entire LaguerreGaussian beam does not intersect the receiver aperture 6010. This issuemay be addressed as illustrated in FIG. 60B. As before the Gaussianwaves 6002 pass through the spiral phase plate 6004 generating LaguerreGaussian waves 6006. Prior to passing through the transmitter aperture6008 the Laguerre Gaussian waves 6006 pass through a pair of lenses6014. The pair of lenses 6014 have an effective focal length 6016 thatfocuses the beam 6018 passing through the transmitter aperture 6008. Dueto the focusing lenses 6014, the focused beam 6018 fully intersects thereceiver aperture 6012. By providing the lenses 6014 separated by aneffective focal length 6016, a more focused beam 6018 may be provided atthe receiver aperture 6012 preventing the loss of data within thetransmission of the Laguerre Gaussian wave 6006.

Referring now to FIG. 61, there is illustrated a simulation modelutilizing a pair of transmitter lenses 6102. Multiple data carrying OAMbeams 6104 at a same wavelength (1550 NM) are multiplexed and passedthrough a pair of transmitter lenses 6102 before being transmitted infree space through the transmitter aperture 6106. The transmitted beam6108 is received at the receiver aperture 6110. The equivalent focallength of these two transmitter lenses 6102 is F₀=F₁ ²/Δ+Δ. Note thatsuch a structure is widely used in traditional free space optic (FSO)systems as a telescope with the output beam of the lenses collimated(i.e., Δ=0), while in the present OAM-based FSO links non-collimatedbeams (i.e., Δ ≠0) are used to enhance system performance.

Since transmitter lenses can focus OAM beams, more signal power may beprovided at a receiver with limited size apertures. FIG. 62A shows thatwhen the equivalent focal length is adjusted to be around thetransmission distance, power loss decreases. Because of fasterdiversions during propagation, higher order OAM beams would benefitsmore from transmitter lenses as shown in FIG. 62B than lower order OAMbeams. In FIG. 62C, there is shown the use of transmitter lenses toreduce power loss in 10 km links where beam divergence causes high powerloss.

As the received beam size could be adjusted by controlling the distancebetween the two transmitter lenses 6102, such as an OAM-based FSO link,would obtain different performance under lateral displacement, receiverangular error and transmitter pointing error. Links in which lateraldisplacement dominates would prefer to have larger receiver beam sizesbecause of the relatively smaller mismatch under the same displacement.Links with receiver angular error would prefer smaller receiver beamsizes and less phase shift would be introduced by the same angular erroras illustrated in FIGS. 63B and 63C. As a combination of displacementand angular error, pointing error would have a trade-off in choosingreceiver beam size. When simulating the OAM +3 as a function of thedistance between two transmitter lenses in a 1 km link and when OAM +1and OAM +3 are transmitted under 8 mm displacement, 8 μrad angular erroror 8 μrad pointing error, which is shown in FIG. 63A. Such a link wouldhave higher SIR (Signal to Interference Ration) under angular error whenA is around 0.25 mm, which refers to an equivalent focal length of about1 km where the receiver beam size is the smallest. Both too large or toosmall a receiver beam size would decrease SIR under pointing error. Notewhen A increases from 0.3 to 0.5 mm, the receiver beam size increases,SIR under displacement is still decreasing because the beam has a largecurvature.

Referring now to FIG. 64 there is illustrated the experimental setup ofan OAM-based FSO link using transmitter lenses. The generated 50 Gbaudsignal 6402 carried by a light wave with a wavelength of 1550 nm isduplicated by a 50/50 coupler 6404 with one beam relatively delayed inan approximately 10 m length of single mode fiber 6405 fordecorrelation. The two fiber branches are coupled to collimators 6406each of which emits a collimated Gaussian beam with a beam diameter of2.2 mm. One beam is converted to OAM +1 by SLM −1 6408, while the otheris converted to a combination of OAM +3 SLM −2 6410. After beingcombined on a beam splitter 6412, the multiplexing OAM beams are splitinto two identical groups at beam splitters 6414 and 6416. One of thecopies of the beam is reflected by three mirrors 6418 to generate OAM −1and OAM −3, which are multiplexed with the other copy of the beam atbeam splitter 6414. The resulting four multiplexed OAM beams are passedthrough two lenses 6420 with focal lengths of 10 cm, the distance ofwhich is adjustable to control the received beam size. After one meterFSO transmission, the beams are sent to SLM 3 loaded receiver 6422 withan inverse spiral phase hologram 6424 of the particular OAM channel tobe detected. Such an OAM beam is converted to a Gaussian light beamwhich is coupled at a coupler 6426 into an SMF (single mode fiber) andsent for coherent detection within a coherent receiver 6428.

FIG. 65 shows a comparison between simulated and experimental power lossof OAM +3 as a function of receiver aperture size when only OAM +3 istransmitted and transmitter and receiver are perfectly aligned. Limitedsize receiver apertures are implemented by adding truncated pattern onto SLM −3. Power loss decreases for greater than 20 DB when receiveraperture size is smaller than 2.5 mm thanks to transmitter lenses withequivalent focal length of approximately 1 m.

FIGS. 66A and 66B show an SIR of OAM +3 when OAM +1 and OAM +3 aretransmitted with angular error and displacement, respectively. Receiverangular errors are introduced to the link by adding tilted phasepatterns on to SLM −3, and lateral displacements are created byadjusting mirror −1 which can laterally shift the beam. SIR of OAM +3with transmitter lenses is 10 to 20 db higher than that of withoutlenses under various angular errors while 5 to 10 db lower underdifferent lateral displacement.

FIGS. 67A and 67B show BER of OAM +3 when OAM +1 and OAM +3 aretransmitted with angular error and displacement respectively. Whenangular error is 100 rad, a link without transmitter lenses could notachieve the 7% overhead forward error correction (FEC) limit of 3.8e-3,while a link with a transmitter lens having an equivalent focal lengthof approximately 1 m could achieve this FEC limit. Moreover, a link withsuch transmitter lenses could still achieve the FEC limit under grad 400angular error with a little power penalty compared with 100 μrad case.On the other hand, such a link would have higher power penalties underdisplacement the link without transmitter lenses.

Details of the above system are further described in Guodong Xie et al.,Performance Metrics and Design Considerations For a Free-Space OpticalOrbital-Angular-Momentum-Multiplexed Communication Link, Vol. 2, No. 4,OPTICA, 357-365 (2015); A. E. Willner et al., Optical CommunicationsUsing Orbital Angular Momentum Beams, ADVANCES IN OPTICS AND PHOTONICS,66-106 (2015); and Long Li et al., Performance Enhancement of anOrbital-Angular-Momentum-Based Free-Space Optical Communication Linkthrough Beam Divergence Controlling, Optical Fiber CommunicationConference (2015) (on file with author), each of which are incorporatedby reference herein in their entirety.

It will be appreciated by those skilled in the art having the benefit ofthis disclosure that this systems and methods for focusing beams withmode division multiplexing provides improved bandwidth and datatransmission capability. It should be understood that the drawings anddetailed description herein are to be regarded in an illustrative ratherthan a restrictive manner, and are not intended to be limiting to theparticular forms and examples disclosed. On the contrary, included areany further modifications, changes, rearrangements, substitutions,alternatives, design choices, and embodiments apparent to those ofordinary skill in the art, without departing from the spirit and scopehereof, as defined by the following claims. Thus, it is intended thatthe following claims be interpreted to embrace all such furthermodifications, changes, rearrangements, substitutions, alternatives,design choices, and embodiments.

What is claimed is:
 1. A method for focusing an orbital angular momentum(OAM) multiplexed beam, comprising: receiving an OAM multiplexed signalfrom a data processing source, the OAM multiplexed signal including aplurality of data streams each having a unique orbital angular momentumapplied thereto and multiplexed together within the OAM multiplexedsignal, each unique orbital angular momentum having a beam helicityvalue greater than l=2; splitting the OAM multiplexed signal into aplurality of OAM multiplexed signals; providing each of the plurality ofOAM multiplexed signals to a transmitting antenna of a plurality oftransmitting antennas of an antenna array; controlling a timing oftransmissions of each of the plurality of OAM multiplexed signals froman associated transmitting antenna of the plurality of transmittingantennas to cause a transmitted plurality of OAM multiplexed signals tofocus at a predetermined focus point located below ground atsubstantially a same time to overcome a divergence of the transmittedplurality of OAM multiplexed signals caused by the beam helicity valueof greater than l=2 for each of the unique orbital angular momentum; andtransmitting each of the plurality of OAM multiplexed signals from anassociated transmitting antenna of the antenna array toward the focuspoint located below the ground as a transmission beam in accordance withthe timing.
 2. The method of claim 1, wherein the step of transmittingfurther comprises transmitting a pulse of a fixed width having at leasta portion of the OAM multiplexed signal therein.
 3. The method of claim2, wherein the step of transmitting further comprises delayingtransmission of a first pulse from a second pulse by a predeterminedamount to cause the first pulse and the second pulse to arrive at thefocus point below the ground at substantially the same time.
 4. Themethod of claim 1, wherein the step of transmitting further comprisestransmitting each of the plurality of OAM multiplexed signals from anassociated transmitting antenna of the antenna array toward a pluralityof focus points located below the ground as a plurality of transmissionbeams.
 5. The method of claim 1 further including: receiving a pluralityof data streams; modulating each of the plurality of data streams;applying an orbital angular momentum twist to each of the plurality ofmodulated data streams; and multiplexing each of the twisted, modulateddata streams into the OAM multiplexed signal.
 6. A system for focusingan orbital angular momentum (OAM) multiplexed beam, comprising: OAMsignal processing circuitry for generating an OAM multiplexed signal,the OAM multiplexed signal including a plurality of data streams eachhaving a unique orbital angular momentum applied thereto and multiplexedtogether within the OAM multiplexed signal, each unique orbital angularmomentum comprising a beam helicity value greater than l=2; a pluralityof antennas comprising an antenna array; an antenna array controlcircuit for controlling transmission of the multiplexed OAM signal fromeach of the plurality of antennas in the antenna array toward a focuspoint located below ground as a transmission beam to cause thetransmitted OAM multiplexed signals to converge at the focus point belowthe ground at substantially a same time to overcome a divergence of thetransmitted plurality of OAM multiplexed signals caused by the beamhelicity value of greater than l=2 for each of the unique orbitalangular momentum.
 7. The system of claim 6, wherein the antenna arraycontrol circuit further transmits at least one pulse of a fixed widthhaving at least a portion of the OAM multiplexed signal therein.
 8. Thesystem of claim 7, wherein the antenna array control circuit furtherdelays transmission of a first pulse from a second pulse by apredetermined amount to cause the first pulse and the second pulse toarrive at the focus point below the ground at substantially the sametime.
 9. The system of claim 6, wherein the antenna array controlcircuit further controls the antenna array to transmit the OAMmultiplexed signal from an associated transmitting antenna of theantenna array toward a plurality of focus points located below theground as a plurality of transmission beams.
 10. The system of claim 6,wherein the OAM processing circuitry further receives a plurality ofdata streams, modulates each of the plurality of data streams, appliesan orbital angular momentum twist to each of the plurality of modulateddata streams and multiplexes each of the twisted, modulated data streamsinto the OAM multiplexed signal.
 11. A controller for an antenna array,comprising: an interface for interconnecting the controller with anantenna array comprising a plurality of transmitting antennas; and aprocessor for generating control signals for controlling transmissionsof an orbital angular momentum multiplexed signal from the antennaarray, the OAM multiplexed signal including a plurality of data streamseach having a unique orbital angular momentum applied thereto andmultiplexed together within the OAM multiplexed signal, each uniqueorbital angular momentum having a beam helicity value greater than l=2,the processor configured to generate control signals to cause theantenna array to transmit the OAM multiplexed signal from each of theplurality of antennas of the antenna array toward a focus point locatedbelow ground as a transmission beam and control a timing of thetransmissions of the OAM multiplexed signal from each of the pluralityof antennas of the antenna array to cause the transmitted OAMmultiplexed signal from each of the plurality of antennas to converge atthe focus point below the ground at substantially a same time toovercome a divergence of the transmitted plurality of OAM multiplexedsignals caused by the beam helicity value of greater than l=2 for eachof the unique orbital angular momentum.
 12. The controller of claim 11,wherein the processor further generates control signals to transmit atleast one pulse of a fixed width having at least a portion of the OAMmultiplexed signal therein.
 13. The controller of claim 12, wherein theprocessor further generates control signals to delay transmission of afirst pulse from a second pulse by a predetermined amount to cause thefirst pulse and the second pulse to arrive at the focus point below theground at substantially the same time.
 14. The controller of claim 11,wherein the processor further generates control signals to control theantenna array to transmit the OAM multiplexed signal from an associatedtransmitting antenna of the antenna array toward a plurality of focuspoints located below the ground as a plurality of transmission beams.15. A method for focusing an orbital angular momentum (OAM) multiplexedbeam, comprising: generating an OAM multiplexed signal, the OAMmultiplexed signal including a plurality of data streams each having aunique orbital angular momentum applied thereto and multiplexed togetherwithin the OAM multiplexed signal, each unique orbital angular momentumcomprising a beam helicity value greater than l=2; transmitting the OAMmultiplexed signal through a transmitter aperture as a transmissionbeam; and focusing the transmission beam on an aperture of a receiverusing a pair of lenses located before the transmitter aperture, the pairof lenses separated by an effective focal length that overcomes adivergence of the transmitted plurality of OAM multiplexed signalscaused by the beam helicity value of greater than l=2 for each of theunique orbital angular momentum and focuses an entire cross-section ofthe transmission beam on the aperture of the receiver.
 16. The method ofclaim 15, wherein the step of focusing further comprises adjustingeffective focal length responsive to transmission distance between thetransmitter and the receiver to decrease power loss.
 17. The method ofclaim 15, wherein the step of focusing further comprises adjustingcross-section of the transmission beam responsive to changes in theeffective focal length.
 18. The method of claim 15, wherein the step offocusing further comprises adjusting effective focal length responsiveto at least one of displacement error, angular error, and pointingerror.
 19. A system for focusing an orbital angular momentum (OAM)multiplexed beam, comprising: OAM signal processing circuitry forgenerating an OAM multiplexed signal, the OAM multiplexed signalincluding a plurality of data streams each having a unique orbitalangular momentum applied thereto and multiplexed together within the OAMmultiplexed signal, each unique orbital angular momentum comprising abeam helicity value greater than l=2; a transmitter including atransmitter aperture for transmitting the OAM multiplexed signal as atransmission beam; a pair of lenses located before the transmitteraperture, the pair of lenses separated by an effective focal length thatovercomes a divergence of the transmitted plurality of OAM multiplexedsignals caused by the beam helicity value of greater than l=2 for eachof the unique orbital angular momentum and focuses an entirecross-section of the transmission beam on an aperture of a receiver. 20.The system of claim 19, wherein the effective focal length is adjustableresponsive to transmission distance between the transmitter and thereceiver to decrease power loss.
 21. The system of claim 19, wherein thecross-section of the transmission beam is adjusted responsive to changesin the effective focal length.
 22. The system of claim 19, wherein theeffective focal length is adjustable responsive to displacement error,angular error, and pointing error.